Review Article On the Nature of Electronic Wave Functions

Review ArticleOn the Nature of Electronic Wave Functions inOne-Dimensional Self-Similar and Quasiperiodic Systems

Enrique Maciaacute

Departamento de Fısica de Materiales Facultad de CC Fısicas Universidad Complutense de Madrid 28040 Madrid Spain

Correspondence should be addressed to Enrique Macia emaciabafisucmes

Received 3 September 2013 Accepted 26 November 2013 Published 30 March 2014

Academic Editors A N Kocharian E Liarokapis and A Oyamada

Copyright copy 2014 Enrique Macia This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The interest in the precise nature of critical states and their role in the physics of aperiodic systems has witnessed a renewed interestin the last few years In this work we present a review on the notion of critical wave functions and in the light of the obtained resultswe suggest the convenience of some conceptual revisions in order to properly describe the relationship between the transportproperties and the wave functions distribution amplitudes for eigen functions belonging to singular continuous spectra related toboth fractal and quasiperiodic distribution of atoms through the space

1 Basic Notions

11 Orderings of Matter The notion of order is one of themost fundamental ones In fact order inspires the besthuman civilization achievements in politics ethics arts andsciences [1] Order pervades also most workings of Nature asthe universe unfolds creating symmetric patterns and stablestructures Among them solid matter arrangements wereinitially categorized in a dichotomist way namely as eitherordered or disordered matter forms In this way orderedmatter was identified with periodic arrays of atoms throughthe three-dimensional space while disordered matter wasrelated to random atomic distributions instead Thus thenotions of crystalline matter and spatial periodicity wereborn interwoven from the very beginning just as amorphousmatter was conceptually related to randomness in a naturalway (Figure 1(a))

Nevertheless the unexpected finding of incommensuratephases during the 1960s and 1970s followed by the discoveryof quasicrystalline alloys in 1982 opened up a discussionforum on the very crystal notion in the crystallographic con-densed matters physics and materials science communitiesIndeed initially it was thought that quasicrystals (short forquasiperiodic crystals) corresponded to a somewhat inter-mediate order form between that of crystals and amorphousmaterials [2] However it was soon realized that quasicrystals(QCs) exhibiting long-range order along with orientational

symmetries not compatible with periodic translations actu-ally represented a new order style which should be properlyinterpreted as a natural extension of the notion of a crystalto structures with quasiperiodic (QP) instead of periodicarrangements of atoms [3] Consequently the InternationalUnion of Crystallography widened in 1992 the very definitionof crystal introducing two separate categories of crystalrepresentatives referred to as periodic and aperiodic crystalsrespectively According to the proposed terms of reference

In the following by ldquocrystalrdquo we mean any solidhaving an essentially discrete diffraction diagramand by ldquoaperiodic crystalrdquo we mean any crystal inwhich three-dimensional lattice periodicity can beconsidered to be absent [4]

Thus QCs along with incommensurate phases belong tothe novel aperiodic crystals category whereas usual periodiccrystals are now known as classical crystals (Figure 1(b))The revamped crystal definition reflects our current under-standing that microscopic periodicity is a sufficient but notnecessary condition for crystallinity Therefore the presenceof a mathematically well-defined long-range atomic ordershould be regarded as the generic attribute of crystallinematter rather than mere periodicity At the same timethe essential attribute of crystallinity is transferred fromreal space to reciprocal space through the recourse to thediffraction patterns hence highlighting the importance of

Hindawi Publishing CorporationISRN Condensed Matter PhysicsVolume 2014 Article ID 165943 35 pageshttpdxdoiorg1011552014165943

2 ISRN Condensed Matter Physics

Disorder Order

Random Periodic

(a) Before 1992

Order

Periodic Aperiodic

(b) After 1992

Amorphous Crystals Classicalcrystals

Aperiodiccrystals

Incommensuratecomposites

QuasicrystalsIncommensuratemodulated

phases

3

44 5 6

gt3

Figure 1 In 1992 the notion of crystal was widened beyondmere periodicity This conceptual diagram presents the positionof aperiodic crystals no longer based on the notion of periodictranslation symmetry among the different orderings of matter Thediverse aperiodic crystal families are arranged according to thedimension 119863 (see (2)) of their embedding hyperspaces (numericallabels)

P

AP

QP

Figure 2 Graphical representation of the hierarchical nestedrelationship among almost periodic (AP) quasiperiodic (QP) andperiodic (P) functions

the Fourier transform in order to properly analyze atomicdensity distributions

Despite the fact that more than two decades have elapsedsince the crystal notion has been properly revisited one canstill find in the literature a lot of works plainly stating thatquasiperiodic systems (QPS) provide an example of inter-mediate structures between ordered and disordered systemsSentences like this certainly rely on a too vague notion of theterm ldquointermediaterdquo which apparently ignores the fact thatevery QP function can be expressed in terms of a numerableset of periodic functions in an appropriate high-dimensionalspace Accordingly periodic functions are but the simplerparticular instances of the more general QP ones From thisperspective QPS are not only perfectly ordered structuresbut theymay even be regarded as having a higher order degreethan periodic ones This viewpoint is nicely illustrated bythe hierarchical relationship between almost periodic (AP)QP and periodic functions shown in Figure 2 Indeed froma mathematical viewpoint periodic functions are a specialcase of QP functions which are in turn a special case of APfunctions1

Therefore rather than adopting the old dichotomist way(which only allows one to get fuzzy qualitative comparisonsas to whether a particular system is less random or moreperiodic than any other one) it may be more fruitful to thinkin terms of the different hierarchies of order to which thesesystems belong (see Figure 7)

Almost periodic functions can be uniformly approxi-mated by Fourier series containing a countable infinity ofpairwise incommensurate reciprocal periods (frequencies)[5 6] When the set of reciprocal periods (frequencies)required can be generated fromafinite-dimensional basis theresulting function is referred to as a QP one2 For the sake ofillustration let us consider an aperiodic crystal whose atomicdistribution is given by a QP function expressed in terms ofits discrete Fourier decomposition

119891 (x) = sum

k119886k119890119894ksdotx

(1)

where the reciprocal vectors are defined by

k =

119873

sum

119895=1

119899119895b119895 (2)

where b119895are reciprocal lattice basis vectors If the minimal

number of these basis vectors is larger than three that is119873 gt 3 in (2) then a higher dimensional description is neededto describe the reciprocal lattice and the related structure isan aperiodic crystal Otherwise we obtain a periodic crystal(Figure 1(b))

The simplest one-dimensional example of a QP functioncan be written as

119891 (119909) = 1198601cos (119909) + 119860

2cos (120572119909) (3)

where 120572 is an irrational number and 1198601and 119860

2are real

numbers It is interesting to note that this QP function can beobtained as a one-dimensional projection of a related periodicfunction in two dimensions

119891 (119909 119910) = 1198601cos119909 + 119860

2cos119910 (4)

through the restriction 119910 = 120572119909 This property is at the basisof the so-called cut and project method which is widely usedin the study of QCs In fact since any QP function can bethought of as deriving from a periodic function in a spaceof higher dimension most of the basic notions of classicalcrystallography can be properly extended to the study of QCsin appropriate hyperspaces [5 7]

12 Extended Localized and Critical Wave Functions Oncewe have clarified that aperiodic crystals do not occupy avague intermediate position between periodic crystal andamorphous matter representatives it is pertinent to indicatethat there exists a physical context in which one can properlytalk about the existence of an intermediate state betweenorder and disorder This scenario is that occurring whena system undergoes a phase transition from solid to liquidstates experiencing critical fluctuations at all scales This

ISRN Condensed Matter Physics 3

0 2583

Sites n

Wav

e fun

ctio

n120595n

(a)

minus233 0 233

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

(b)

minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

(c)

Figure 3 Representative wave function amplitudes distributions corresponding to (a) an extended state (b) an exponentially localized stateand (c) a critical state (adapted from [8] with permission from Ostlund and Pandit copy 1984 by the American Physical Society)

situation is referred to as a passage through a ldquocriticalpointrdquo At the critical temperature various thermodynamicfunctions develop a singular behavior which is related tolong-range correlations and large fluctuations Actually thesystem should appear identical on all length scales at exactlythe critical temperature and consequently it would be scaleinvariant All these features characteristic of thermodynamicphase transitions have been progressively incorporated tothe study of both incommensurate and QPS since analogoustransitions can occur in a solid that preserves its structuralintegrity but experiences a transition from a metallic-likebehavior to an insulator-like one for instance This kind of

phase transitions affecting the transport properties ratherthan the lattice structure of a givenmaterial is very importantto us since the metal-insulator transition provides the basicgrounds necessary to introduce some fundamental notionsand the related terminology

Indeed the metallic regime is understood in terms ofconducting extended electronic wave functions propagatingthrough the solid (Figure 3(a)) whereas the insulating regimeis explained in terms of decaying wave functions correspond-ing to states localized close to the lattice atoms (Figure 3(b))During the metal-insulator transition the electronic wavefunctions experience substantial changes exhibiting a rather


0

x

uk(x)

minusa2 a2

(a)

x

minus5a 0 5a

eikxuk(x)

(b)

Figure 4 Illustration of a Bloch function In (a) we show the periodic function 119906119896(119909) centered at the origin of the unit cell within the range

minus1198862 le 119909 le 1198862 where 119886 is the lattice constant In (b) the Bloch function is constructed by using the function shown in (a) At every latticesite (solid circles representing atoms) the function 119906

119896is modulated by the plane wave 119890119894119896119909 (only the real part is plotted) (Courtesy of Uichiro

Mizutani from [9] by permission of Cambridge University Press)

involved oscillatory behavior and displaying strong spatialfluctuations at different scales (Figure 3(c)) Due to thispeculiar spatial distribution of their amplitudes (reminiscentof the atomic distribution observed in materials undergoinga structural phase transition at the critical point) theseelectronic states are referred to as critical wave functions

In order to properly appreciate the main characteristicfeatures of critical states let us recall first the explicit mathe-matical expressions for extended and localized states It is wellknown that in periodic crystals extended states are describedin terms of the so-called Bloch functions The conceptualappeal of Bloch functions in the description of the physicalproperties of classical crystals is easily grasped by solving theSchrodinger equation describing the motion of an electronwith awave function120595 energy119864 and effectivemass119898 underthe action of a potential 119881(119909) in one dimension

ℎ2

2119898

1198892120595

1198891199092+ [119864 minus 119881 (119909)] 120595 = 0 (5)

where ℎ is the reduced Planckrsquos constant In the absence ofany interaction (ie119881(119909) = 0 for all 119909) the solution to (5) fora free electron is readily obtained as a linear combination ofplanewaves of the form120595

plusmn(119909) = 119860

plusmn119890plusmn119894119896119909 where 119896 = radic2119898119864ℎ

is the wave vector The next step is to consider the motionof an electron interacting with the atoms forming a crystallattice with a lattice constant 119886 Since this lattice is periodic ina classical crystal the resulting interaction potential naturallyinherits the periodicity of the lattice so that one has 119881(119909 +

119899119886) = 119881(119909) where 119899 isin Z Within this context the celebratedBlochrsquos theorem states that the solution to (5) now reads

120595 (119909) = 119906119896 (119909) 119890

119894119896119909 (6)

where the function 119906119896(119909) is real and periodic with the same

period than that of the lattice that is 119906119896(119909 + 119899119886) = 119906

119896(119909) for

all 119899 isin Z In addition the 119906119896(119909) function generally depends

on the electron wave vector which can take certain valuescomprised within a series of allowed intervals minus119898120587119886 le

119896 le 119898120587119886 119898 isin N which ultimately define the electronic

energy spectrumTherefore the periodicity of function 119906119896(119909)

guarantees the periodicity of the Bloch function itself for

120595 (119909 + 119899119886) = 119906119896 (119909 + 119899119886) 119890

119894119896(119909+119899119886)

= 119906119896 (119909) 119890

119894119896119909119890119894119896119899119886

= 119906119896 (119909) 119890

119894119896119909= 120595 (119909)

(7)

It is important to note that the 119906119896(119909) function usually

describes the structure of the wave function in the atomsneighborhood and it is generally relatively localized aroundthem (Figure 4(a)) Thus the extended nature of Blochfunctions ultimately arises from the plane wave modulationas it is illustrated in Figure 4(b)

On the other hand in amorphousmaterials characterizedby a random distribution of atoms through the space theelectronic states are exponentially localized according to anexpression of the form

120595119899 (119909) = 119860

119899119890minusℓ119899|119909| (8)

where 119899 labels the lattice position and ℓminus1

119899provides a measure

of the spatial extension of thewave function which is referredto as its localization length At this point it is important toemphasize that the ultimate reason leading to the localizationof electronic states in random chains is not the presenceof exponentially decaying modulations in (8) but the factthat both the amplitudes 119860

119899and the reciprocal localization

lengths ℓ119899form a random ensemble [10] This property

guarantees that possible resonances between electronic statesbelonging to neighboring atoms cannot extend to other atomslocated far away along the chain In fact as soon as a shortrange correlation is present in an otherwise disordered chainone can observe the emergence of a significant number ofrelatively extended states [11 12]

In summary Bloch states are the prototypical states ofperiodic systems whereas exponentially localized states arethe typical states found in random systems3 Accordinglythe states occurring at the critical point in a metal-insulatortransition that is critical states were originally defined asbeing neither Bloch functions nor exponentially localized


states but occupying a fuzzy intermediate position betweenthem

These states which we will term critical have amaximum at a site (in the lattice) and a series ofsubsidiary maxima at (a number of other) siteswhich do not decay to zero [13]

As we previously mentioned the term ldquocriticalrdquo was orig-inally borrowed from thermodynamics where it has usuallybeen applied to describe a conventional phase transitionwhere a state undergoes fluctuations in certain physicalproperties which are the same on all length scales Followinga chronological order the concept of critical wave functionwas born in the study of the Anderson Hamiltonian whichdescribes a regular lattice with site-diagonal disorder Thismodel is known to have extended states for weak disorderin 3D systems as well as in 2D samples with a strongmagnetic field For strong disorder on the other hand theelectronic states are localized For 1D systems it was provedthat localized states decay exponentially in space in mostcases [14] However this exponential decay relates to theasymptotic evolution of the envelope of the wave functionwhile the short-range behavior is characterized by strongfluctuationsThemagnitude of these fluctuations seems to berelated to certain physical parameters such as the degree ofdisorder which in turn controls the appearance of the so-called mobility edges Approaching a mobility edge from theinsulator regime the exponential decay constant diverges sothat the wave function amplitudes can be expected to featurefluctuations on all length scales larger than the lattice spacingThis singular fact turns out to be very convenient to explainmetal-insulator transitions

Thus the notion of ldquocriticalityrdquo can be understood asfollows An extended wave function is expected to extendhomogeneously over the whole sample On the other hand forawave function localized at a particular site of the sample oneexpects its probability density to display a single dominantmaximum at or around this site and its envelope func-tion is generally observed to decay exponentially in spaceOn the contrary a critical state is characterized by strongspatial fluctuations of the wave function amplitudes Thisunusual behavior consisting of an alternatively decaying andrecovering of the wave function amplitudes is illustrated inFigure 5 Two main features of this wave function amplitudedistribution must be highlighted On the one hand althoughthe main local maxima are modulated by an overall decayingenvelope this envelope cannot be fitted to an exponentialfunction On the other hand the subsidiary peaks around themain local maxima display self-similar scaling features

13 Incommensurate and Self-Similar Systems The discoveryof QCs spurred the interest in the study of specific QP latticemodels describing the electron dynamics in one-dimensionalQPS As a first step most studies made use of both theelectron independent and the tight-binding approximations4by considering a discretized version of the time-independentSchrodinger equation (5) given by

119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= (119864 minus 119881119899) 120595119899 (9)

Table 1 Functions determining the potential energy values119881119899in (9)

for different incommensurate models considered in the literaturewhere 120582 gt 0 is the amplitude of the potential ] is an arbitrary phase119909(119899) equiv 119899 + 120590[119899120590] 120590 = (radic5 minus 1)2 is the inverse of the golden meanand 119887 is a real number

Model PotentialAubry and Andre 120582 cos(119899120572 + ])Bichromatic 120582

1cos(119899 + ]) + 120582

2cos(119899120572 + ])

Soukoulis and Economou 120582 cos(119899120572 + ]) + 1205821015840 cos 2(119899120572 + ])

Maryland 120582 tan(119899120572 + ])Kim 120582 cos(119909(119899)120572 + ])Hiramoto 120582 tanh[119887 cos(119899120572 + ])] tanh 119887

where 120595119899stands for the amplitude of the wave function in

the 119899th lattice site of the chain 119881119899are the on-site energies

(accounting for the atomic potentials) at that site 119905119899119899plusmn1

are thecorresponding transfer integrals (accounting for the hoppingof the electron between neighboring atoms) and 119864 is theenergy of the state5 In the first place we note that thisequation reduces to well-known systems of physical interestin certain particular cases For instance if 119881

119899andor 119905

119899119899plusmn1

are uncorrelated random variables with uniform probabilitydistribution (9) describes a disordered system within theso-called Anderson model On the contrary if the 119881

119899and

119905119899119899plusmn1

parameters obey a periodic sequence we will be dealingwith a classical periodic crystal Therefore (9) allows fora unified mathematical treatment encompassing periodicrandom and QPS

In order to specify a given QPS one must indicate its on-site energies 119881

119899 and transfer integrals 119905

119899119899+1 sequences

Potentials usually considered in (9) can be classified intotwo broad families namely incommensurate and self-similarpotentials

(i) Incommensurate potentials are characterized by thepresence of (at least) two superimposed periodicstructures whose corresponding periods are incom-mensurate The origin of incommensurability may bestructural (as it occurs when two different periodicsublattices form a whole system) or dynamical whenone of the periodicities is associated with the crystalstructure and the other one is related to the behaviorof elementary excitations propagating through thesolid (see Section 15)

(ii) Self-similar potentials on the contrary describe anew ordering of matter based on the presence ofinflation symmetries replacing the translation ones

In the case of incommensurate systems the potentialamplitude is given by a periodic function whose argumentdepends on 119899120572 120572 being an irrational number Some of themost representative models studied up to now are given inTable 1

On the other side in most self-similar systems studied todate the aperiodic sequence 119881

119899is defined in terms of certain

substitutional sequences A substitution sequence is formallydefined by its action on an alphabetA = 119860 119861 119862 whichconsists of certain number of letters generally corresponding


minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0 987

Sites n

Sites n

minus987

minus233 0 233

(a)

(b)

(c)

Figure 5 Illustration of a critical wave function The amplitude distribution shown in (a) for the wave function as a whole is identicallyreproduced at progressively smaller scales (indicated by the arrows) in (b) and (c) after a proper rescaling This is a distinctive feature of itscharacteristic self-similarity property This wave function corresponds to the eigenstate located at the center of the spectrum 119864 = 0 in theAubry-Andre model (see Section 15) with a potential strength 120582 = 2 (adapted from [8] with permission from Ostlund and Pandit copy 1984 bythe American Physical Society)

to different kinds of atoms in actual realizationsThe substitu-tion rule starts by replacing each letter by a finite word as it isillustrated in Table 2 The corresponding aperiodic sequenceis then obtained by iterating the substitution rule 119892 startingfrom a given letter of the setA in order to obtain a QP stringof letters

For instance the Fibonacci sequence is obtained from thecontinued process 119860 rarr 119860119861 rarr 119860119861119860 rarr 119860119861119860119860119861 rarr

119860119861119860119860119861119860119861119860 rarr sdot sdot sdot Another popular sequence is theso-called Thue-Morse sequence which has been extensivelystudied in the mathematical literature as the prototype of asequence generated by substitution In this case the continuedprocess reads 119860 rarr 119860119861 rarr 119860119861119861119860 rarr 119860119861119861119860119861119860119860119861 rarr

119860119861119861119860119861119860119860119861119861119860119860119861119860119861119861119860 rarr sdot sdot sdot The number of letters

in this sequence increases geometrically 119873 = 2119896 where 119896

indicates the iteration order In the infinite limit the relativefrequency of both kinds of letters in the sequence takes thesame value that is 120592

119860= 120592119861

= 12 This result contrastswith that corresponding to the Fibonacci sequence where120592119860= 120591minus1 and 120592

119861= 120591minus2 with 120591 = (1 + radic5)2 being the golden

mean Another important difference is that in the Fibonaccisequence 119861 letters always appear isolated whereas in Thue-Morse sequence both dimers 119860119860 and 119861119861 appear alike

We note that QPS based on substitution sequence relatedalphabets take on just a few possible values (say twofor Fibonacci and Thue-Morse or four for Rudin-Shapirosequences) whereas QPS based on the discretization ofcontinuous potentials (eg 119881

119899= 120582 cos(119899120572 + ])) can take on


Table 2 Substitution rules most widely considered in the study ofself-similar systems where 119899 and119898 are positive integers

Sequence SetA Substitution rule

Fibonacci 119860 119861119892 (119860) = 119860119861

119892(119861) = 119860

Silver mean 119860 119861119892(119860) = 119860119860119861

119892(119861) = 119860

Bronze mean 119860 119861119892(119860) = 119860119860119860119861

119892(119861) = 119860

Precious means 119860 119861119892(119860) = 119860

119899119861

119892(119861) = 119860

Copper mean 119860 119861119892(119860) = 119860119861119861

119892(119861) = 119860

Nickel mean 119860 119861119892(119860) = 119860119861119861119861

119892(119861) = 119860

Metallic means 119860 119861119892(119860) = 119860119861

119899

119892(119861) = 119860

Mixed means 119860 119861119892(119860) = 119860

119899119861119898

119892(119861) = 119860

Thue-Morse 119860 119861119892(119860) = 119860119861

119892(119861) = 119861119860

Period-doubling 119860 119861119892(119860) = 119860119861

119892(119861) = 119860119860

Ternary Fibonacci 119860 119861 119862

119892(119860) = 119860119862

119892(119861) = 119860

119892(119862) = 119861

Rudin-Shapiro 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119860119862

119892(119862) = 119863119861

119892(119863) = 119863119862

Paper folding 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119862119861

119892(119862) = 119860119863

119892(119863) = 119862119863

a significantly larger set of values This property makes thesepotentials more complex from Shanonrsquos entropy viewpoint[15]

14 Spectral Measure Classification A key question in anygeneral theory of QPS regards the relationship betweentheir atomic topological order determined by a given QPdistribution of atoms and bonds and the physical propertiesstemming from that structure At the time being a generaltheory describing such a relationship is still lacking Thisunsatisfactory situation has considerably spurred the interestin studying the main properties of QP Schrodinger operatorsfrom amathematical perspective To this end it is convenientto arrange (5) in the following form6

[minus1198892

1198891199092+ 119881 (119909)]120595 = 119864120595 (10)

which can be regarded as an eigenvalue equation involvingthe Schrodinger operator within the square bracket Withinthis framework the nature of an (eigen)state is determined bythe measure of the spectrum to which it belongs

Most rigorous mathematical results in the field have beenderived from the study of nearest-neighbor tight-bindingmodels described in terms of a convenient discretization of(10) given by the Hamiltonian7

(119867120595)119899= 119905120595119899+1

+ 119905120595119899minus1

+ 120582120599 (119899) 120595119899 119899 isin Z (11)

where 120582 gt 0 measures the potential strength 120599(119899) = [(119899 +

1)120572] minus [119899120572] where [119909] denotes the integer part of 119909 and120572 is generally the golden mean or its reciprocal From amathematical point of view these models belong to the classof AP Schrodinger operators which display unusual spectralproperties

Indeed according to Lebesguersquos decomposition theoremthe energy spectrum of any measure in R119899 can be uniquelydecomposed in terms of just three kinds of spectral measures(and mixtures of them) namely pure-point (120583

119875) absolutely

continuous (120583AC) and singularly continuous (120583SC) spectra inthe following form

120583 = 120583119875cup 120583AC cup 120583SC (12)

Suitable examples of physical systems containing both thepure-point andor the absolutely continuous components intheir energy spectra are well known with the hydrogenicatom being a paradigmatic instance On the other hand theabsence of actual physical systems exhibiting the singularcontinuous component relegated this measure as a merelymathematical issue for some time From this perspective thediscovery of QCs bridged the long standing gap between thetheory of spectral operators in Hilbert spaces and condensedmatter theory [16 17]

Now from the viewpoint of condensed matter physicsthere are two different (though closely related) measuresone can consider when studying the properties of solidmaterials On the one hand we have the measure related tothe atomic density distribution which determines the spatialstructure of the solidOn the other hand we have themeasurerelated to the energy spectra of the system which describesthe electronic structure (or the frequency distribution ofatomic vibrations in the case of the phonon spectrum) andits related physical properties In order to characterize thesolid structure it is convenient to focus on the nature ofthe measure associated with the lattice Fourier transformwhich is related to the main features of X-ray electron orneutron diffraction patterns For the sake of illustration inFigure 6 we show the Fourier amplitude distributions forthree representative QPS [18]

Since the electronic structure of a system is ultimatelyrelated to the spatial distribution of its constituent atomsthroughout the space and to their bonding properties oneexpects a close relation to exist between the electronic energyspectral measure and the Fourier lattice measure In thisregard a particularly relevant result obtained from the studyof QPS is the so-called gap-labeling theorem which providesa relationship between reciprocal space (Fourier) spectra andHamiltonian energy spectra In fact this theorem relates theposition of a number of gaps in the electronic energy spectrato the singularities of the Fourier transform of the substratelattice [19ndash21] Accordingly in order to gain additional insight


0 20 40 60 80 100

Fibonacci 120575-like Bragg peaks

0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()

Singular continuousThue-Morse

0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50

Absolutely continuousPseudorandom

Figure 6 Absolute value of the Fourier coefficients of a quasiperi-odic (Fibonacci) structure an aperiodic Thue-Morse structure witha singular-continuous spectrum and an aperiodic Rudin-Shapirostructure with an absolutely continuous spectrum (Courtesy of LucaDal Negro)

into the relationship between the type of structural orderpresent in an aperiodic solid (as determined by its latticeFourier transform) and its related transport properties (asdetermined by the main features of the energy spectrum andthe nature of its eigenstates) it is convenient to introduce thechart depicted in Figure 7 In this chart we present a classifica-tion scheme of aperiodic systems based on the nature of theirdiffraction spectra (in abscissas) and their energy spectra (inordinates) In this way we clearly see that the simple classifi-cation scheme based on the periodic-amorphous dichotomyis replaced by a much richer one including nine differententries In the upper left corner we have the usual periodiccrystals exhibiting pure-point Fourier spectra (well-definedBragg diffraction peaks) and an absolutely continuous energyspectrum (Bloch wave functions in allowed bands) In thelower right corner we have amorphous matter exhibiting anabsolutely continuous Fourier spectrum (diffuse spectra) anda pure-point energy spectrum (exponentially localized wavefunctions) By inspecting this chart one realizes that althoughFibonacci and Thue-Morse lattices share the same kind ofenergy spectrum (a purely singular continuous one) theyhave different lattice Fourier transforms so that these QPSmust be properly classified into separate categories

At the time being the nature of the energy spectrumcorresponding to the Rudin-Shapiro lattice is yet an openquestion Numerical studies suggested that some electronic

Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling

Thue-Morse Rudin-Shapiro

Ideal quasicrystal

Amorphous matter

Lattice Fourier transform

120583P 120583AC120583SC

120583P

120583AC

120583SC

Figure 7 Classification of aperiodic systems attending to thespectral measures of their lattice Fourier transform and theirHamiltonian spectrum energy (from [22] Macia with permissionfrom IOP Publishing Ltd)

states are localized in these lattices in such a way that the rateof spatial decay of the wave functions is intermediate betweenpower and exponential laws [23ndash25] These results clearlyillustrate that there is not any simple relation between thespectral nature of the Hamiltonian describing the dynamicsof elementary excitations propagating through an aperiodiclattice and the spatial structure of the lattice potential

In the light of these results it is tempting to establish aone-to-one correspondence between extended localized andcritical states introduced in Section 12 on the one handand the three possible spectral measures namely absolutelycontinuous pure-point and singular continuous spectra onthe other hand Indeed the study of several physical systemsprovided evidence on the correspondence between extendedstates and absolutely continuous spectra as well as betweenexponentially localized states and pure-point measures Forinstance periodic lattices described in terms of the Kronig-Penney model have a mixed spectrum consisting of a pure-point component for low energy values and an absolutelycontinuous component for higher enough energies8

What about singular continuous spectra The energyspectrum of most QPS considered to date seems to be apurely singular continuous one which is supported on aCantor set of zero Lebesgue measure Thus the spectrumexhibits an infinity of gaps and the total bandwidth of theallowed states vanishes in the thermodynamic limit Thoughthis property has only been proven rigorously for QPS based


on the Fibonacci [26ndash28] Thue-Morse and period-doublingsequences [29 30] it is generally assumed that it may be aquite common feature of most QPS [31] and it has become arelatively common practice to refer to their states genericallyas critical states on this basis However this semantics doesnot necessarily imply that all these critical states whichbelong to quite different QPS in the spectral chart shown inFigure 7 will behave in exactly the same way from a physicalviewpointThis naturally leads to somemisleading situations

In order to clarify this issue one should start by addressingthe following questions

(i) What is the best term to refer to the eigenstatesbelonging to a singular continuous spectrum

(ii) What are the specific features (if any) of these statesas compared to those belonging to absolutely contin-uous or pure-point spectra

(iii) What are the characteristic physical properties (if any)of states belonging to singular continuous spectra

(iv) To what extent are these properties different fromthose exhibited by Bloch states and exponentiallylocalized states respectively

To properly answer the above questions one may rea-sonably expect that the study of a system where the threepossible kinds of wave functions were simultaneously presentmay shed some light on the physical nature of critical wavefunctions and their main differences with respect to bothBloch and exponentially localized functions

15 The Aubry-Andre Model In 1980 Aubry and Andrepredicted that for a certain class of one-dimensional QPSa metal-insulator localization phase transition can occur[32] Below the transition all the states of the system areextended whereas above the critical point all states arelocalized At exactly the critical point all the wave functionsbecome critical ones Therefore the Aubry-Andre modelprovides an illustrative example of a QPS which can exhibitextended localized or critical wave functions depending onthe value of a control parameter whichmeasures the potentialstrength Accordingly this parameter can be regarded as anorder parameter controlling the existence of ametal-insulatortransition in the system

Explicitly the Schrodinger equation for the Aubry-Andremodel is given by

[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)

where 120572 is an irrational number 120582 is a real number modulat-ing the potential strength and ] is a real number describing aphase shiftThere are twowell-known physical systemswhichcan be described by (13)The first is the motion of an electronin a two-dimensional square lattice with lattice constant 119886in the presence of a magnetic field H perpendicular to theplane In this case 120572 = 119890119886

2119867(ℎ119888) is related to the ratio of

the magnetic flux through the lattice unit cell (1198862119867) to thequantum magnetic flux (ℎ119888119890) where 119888 is the speed of light

and 119890 is the electron charge The second example is providedby the one-dimensional electron dynamics in an aperiodiccrystal characterized by the presence of two superimposedperiodic potentials the main one of period 119871

1 determines

the position of the discrete lattice points and the subsidiaryone of period 119871

2 describes a displacive modulation of the

structureWithin the tight-binding approach and the nearest-

neighbor approximation it is convenient to discretize (13) inthe form9

120595119899minus1

+ 120595119899+1

+ [119864 minus 120582 cos (2120587119899120572 + ])] 120595119899 = 0 (14)where 119899 labels the lattice sites and the lattice constant definesthe length scale (119886 equiv 1) Considered as an operator inℓ2(Z) (14) describes a bounded and self-adjoint operator

For a rational value of 120572 (periodic crystal) (14) can be solvedby applying Blochrsquos theorem and the energy spectrum isabsolutely continuous For irrational values of 120572 (aperiodiccrystal) the nature of the energy spectrum depends on thevalue of the potential strength 120582 For 120582 lt 2 the spectrum isabsolutely continuous with extended Bloch wave functions(Figure 8) when 120582 gt 2 the spectrum is pure-point andcontains exponentially localized states At 120582 = 2 the spectrumis singular continuous and all the wave functions becomecritical (Figure 5)10

The fact that when the critical potential strength 120582 =

2 is adopted the metal-insulator transition occurs for allthe eigenstates independently of their energy was rathersurprising Indeed previous experiencewith random systemsshowed that the states closer to the band edges become moreeasily localized than those located at the band center It is nowunderstood that the simultaneous change of the localizationdegree for all the eigenstates is a unique property of the so-called self-dual systems of which the Aubry-Andre model isa typical representative The self-dual property expresses thefollowing symmetry if we substitute

120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)

in (14) we obtain the so-called dual representation

119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)

where the dual variables are

=4

120582 119864 =

2119864

120582 ] = 119896 (17)

By comparing (14) and (16) we see that the Fouriercoefficients 119886

119898satisfy the same eigenvalue equation as the

wave functions amplitudes 120595119899when 120582 = 2 In this case it is

said that (14) and (16) are self-dual (remain invariant undera Fourier transformation) Accordingly if the eigenstate 120595

119899

is spatially localized then the eigenstate of the dual problem119886119898 is spatially extended and vice versaThedegree of localization of theAubry-Andremodel states

can be collectively characterized by an exponent 120573 defined as[8]

120573 equiv lim119898rarrinfin

[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)

Figure 8 Illustration of a Bloch function in the weak potential regime of the Aubry-Andre model The amplitude distribution shown in (a)for the wave function as a whole can be expressed in the form 120595

119899= 1198901198941198991205872

119906(119909) where the hull periodic function 119906(119909) = 119906(119909 + 1) is shown in(b) This wave function corresponds to the eigenstate located at the center of the spectrum 119864 = 0 with a potential strength 120582 = 15 (adaptedfrom [8] with permission from Ostlund and Pandit copy 1984 by the American Physical Society)

where 119901 is some integer (whichmay depend on 119896) describingthe lattice scaling properties 119902

119898is the denominator of suc-

cessive rational approximants to 12057211 and |120595119899|2 is the squared

wave function amplitude From the above definition it followsthat 120573 = 0 for extended states and 120573 = minus1 for exponentiallydecaying localized states In their numerical study Ostlundand Pandit considered the Aubry-Andre model with 120572 =

120590 = (radic5 minus 1)2 the reciprocal of the golden mean andthey focused on the eigenstate corresponding to the energy119864 = 0 (located at the spectrum center) By assuming 119901 = 3 (assuggested by the lattice scaling of the critical wave functionsshown in Figure 5) they found

120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)

Thus one obtains Bloch functions when 120582 lt 2 (Figure 8)and exponentially localized functions when 120582 gt 2 whoselocalization length depends on the adopted potential strengthaccording to the relationship ℓ

minus1sim |120582 minus 2| (Figure 9) The

value of the exponent 120573 obtained for 120582 = 2 clearly indicatesthe states are neither exponentially localized nor Bloch-likeextended since minus1 lt 120573 lt 0 in this case From a closerinspection of the wave function amplitudes distributionplotted in Figure 5 they concluded that the structure aroundthe main local peaks approaches a length-rescaled version ofthe structure around 119899 = 0 peak Let 119899

119896be a label denoting

the location of the subsidiary peaks around the 119899th mainlocal peak then their corresponding amplitudes are related to120595119899119896+119899

= 120589120595119899 where 120589 = 0315 plusmn 0005 is a scaling factor given

by a constant fraction of the 119899 = 0 peak Therefore there isa whole hierarchy of subsidiary peaks with peak height 120589119898 ofthe central 119899 = 0 peakThe self-similar behavior of the criticalwave function implies that strictly speaking they cannot beregarded as localized since the wave function amplitudesnever decay to zero although the points exhibiting largeamplitudes are further and further apart from each other

It is important to highlight that the self-similar distri-bution of the wave function amplitudes is not restricted tothe critical potential strength value 120582 = 2 In Figure 9 themagnitude of the wave function at the band center is shownat the supercritical regime 120582 = 205 By comparing Figures5 and 9 we see that for potential strengths relatively closeto the critical one but certainly located in the supercriticalregime the wave functions envelope decays more rapidlyhence leading to amore pronounced confinement of the wavefunction support around the central site 119899 = 0 as expectedfor eigenstates undergoing a transition to an exponentiallylocalized behavior But this envelope is still encapsulating aself-similar amplitudes pattern distribution

It was early suggested by Aoki that critical wave func-tions in the Aubry-Andre model may be characterized bysome fractal dimensionality [33 34] Later on Soukoulisand Economou [35] numerically demonstrated the fractalcharacter of certain eigenfunctions in disordered systemsand characterized their amplitude behavior by a fractaldimensionality What is more interesting is that the fractalcharacter of the wave function itself was suggested as anew method for finding mobility edges The observation ofanomalous scaling of both the moments of the probabilitydistribution and the participation ratio near the localization


minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)

Figure 9 Illustration of a localized critical wave function The amplitude distribution shown in (a) for the wave function as a whole isidentically reproduced at progressively smaller scales in (b) and (c) after a proper rescaling This is a distinctive feature of its characteristicself-similarity property (This wave function corresponds to the eigenstate located at the center of the spectrum 119864 = 0 in the Aubry-Andremodel with a potential strength 120582 = 205) (adapted from [8] with permission from Ostlund and Pandit copy 1984 by the American PhysicalSociety)

threshold in the Anderson model strongly suggested that acritical wave function cannot be adequately treated as simplyfractal [36] Instead a multifractal measure is characterizedby a continuous set of scaling indices 119886 and fractal dimensions119891(119886) Accordingly the wave functions cannot be describedas homogeneous fractals [35 37ndash40] For an extended wavefunction one can obtain a single point 119891 = 119886 = 1 whichexpresses the absence of self-similar features in the wavefunction amplitudes distribution When a wave function islocalized the 119891(119886) spectrum consists of two points (if thechain length is longer than the localization length) one being119891(0) = 0 and the other being 119891(infin) = 1 For a criticalwave function one gets a continuous 119891(119886) spectrum but anon-self-similar wave function shows quite different shapesin each scale and does not yield a119891(119886) spectrum independentof the systems size Thus the self-similarity of a critical wavefunction is characterized by the size independence of its 119891(119886)spectrum

16 Fractal Energy Spectra In his pioneering article Hofs-tadter put forward the following fundamental question whatis the meaning (if any) of a physical magnitude whose very

existence depends on the rational or irrational nature of thenumbers in terms of which this magnitude is expressed

Common sense tells us that there can be no physi-cal effect stemming from the irrationality of someparameter because an arbitrarily small change inthat parameter would make it rationalmdashand thiswould create some physical effect with the prop-erty of being everywhere discontinuous which isunreasonable [41]

To further analyze this question in his study of theHarperrsquos equation12

120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)

he considered the dependence of the spectrum Lebesguemeasure (see Section 13) as a function of the 120572 param-eter value and concluded that the measure has a verypeculiar behavior at rational values of 120572 the measure isdiscontinuous13 since there are irrationals arbitrarily nearany rational yet at irrational 120572 values the measure iscontinuous This property is ultimately related to the highlyfragmented nature of the energy spectrum for irrational 120572


Figure 10 Harperrsquos model energy spectrum for irrational 120572 valuesThe energy is in abscissas ranging between 119864 = plusmn4 The fractionalpart of 120572 ranging from 0 to 1 is plotted in the ordinate axis (from[41] with permission from Hofstadter copy 1976 by the AmericanPhysical Society)

values observed in his celebrated butterfly spectra shown inFigure 10

This graph has some very unusual propertiesThelarge gaps form a very striking pattern somewhatresembling a butterfly perhaps equally strikingare the delicacy and beauty of the fine-grainedstructureThese are due to a very intricate schemeby which bands cluster into groups which them-selves may cluster into larger groups and so on[41]14

Very similar features have been reported by differentauthors from the study of different QPS energy spectra (seeSection 21) and can be summarized as follows

(i) The energy spectrum of most self-similar systemsexhibits an infinity of gaps and the total bandwidth ofthe allowed states vanishes in the119873 rarr infin limitThishas been proven rigorously for systems based on theFibonacci [27 28]Thue-Morse and period-doublingsequences [27]

(ii) The position of the gaps can be precisely determinedthrough the gap labeling theorem in some definitecountable set of numbers [19ndash21]

(iii) Scaling properties of the energy spectrum can bedescribed using the formalism of multifractal geom-etry [42 43]

An illustrative example of the spectrum structure cor-responding to two QPS is shown in Figures 11 and 12 Byinspecting these figures we clearly appreciate the followingprefractal signatures

(i) the spectra exhibit a highly fragmented structuregenerally constituted by as many fragments as thenumber of atoms present in the chain

(ii) the energy levels appear in subbands which concen-trate a high number of states and which are separatedby relatively wide forbidden intervals

(iii) the degree of internal structure inside each subbanddepends on the total length of the chain and thelonger the chain the finer the structure which dis-plays distinctive features of a self-similar distributionof levels

Taken altogether these features provide compelling evi-dence about the intrinsic fractal nature of the numericallyobtained spectra which will eventually show up with math-ematical accuracy in the thermodynamic limit 119873 rarr infinNow as one approaches this limit it is legitimate to questionwhether a given energy value actually belongs to the energyspectrum Indeed this is not a trivial issue in the case ofhighly fragmented spectra supported by a Cantor set of zeroLebesgue measure and can be only guaranteed on the basisof exact analytical results In fact because of the presenceof extremely narrow bands special care is required in orderto avoid studying states belonging to a gap and erroneouslyinterpreting their features as those proper critical states

In this regard it should also be noted that irrationalnumbers cannot be explicitly included in a computing codeas that but only in terms of approximate truncated decimalexpressions Accordingly one must very carefully check thatthe obtained results are not appreciably affected by the trun-cation To this end one should consider the systematic useof successive approximants of an irrational number in orderto explore the possible influence of its irrational character(if any) in the physical model under study In fact one canimplement numerically an empirical scaling analysis inwhichthe QPS is approximated by a sequence of periodic systemswith progressively larger unit cells of size 119902

119899defined by the

optimal rational approximants to 120572 namely 120572119899

= 119901119899119902119899

In this way by checking that finer discretization producesalmost the same results one can be confident enough of thereliability of the obtained results [44]

2 One-Dimensional Aperiodic Systems

Broadly speaking an obvious motivation for the recourseto one-dimensional (1D) models in solid state physics is thecomplexity of the full-fledged problem In the particular caseof quasicrystalline matter this general motivation is furtherstrengthened by the lack of translational symmetry thoughthe presence of a well-defined long-range orientational orderin the system also prevents a naive application of proceduresspecifically developed for the study of random structures inthis case

We can also invoke more fundamental reasons support-ing the use of 1D models as a first approximation to thestudy of realistic QPS In fact most characteristic features ofQPS like the fractal structures of their energy spectra andrelated eigenstates can be explained in terms of resonantcoupling effects in the light of Conwayrsquos theorem Thereforethe physical mechanisms at work are not so dependent on thedimension of the system but are mainly determined by theself-similarity of the underlying structure [46] Consequently


1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)

Figure 11 Fragmentation pattern of the energy spectrum of a trans-polyacetylene quasiperiodic chainThenumber of allowed subbandsincreases as a function of the system size expressed in terms of theFibonacci order 119899 as 119873 = 119865

119899(from [45] with permission from

Elsevier)

the recourse to 1D models can be considered as a promisingstarting point for such models encompass in the simplestpossible manner most of the novel physics attributable to theQP order

21 Quasiperiodic Binary Alloys Several representativeexamples of binary Fibonacci chains composed of two typesof atoms say A and B which have been profusely studied inthe literature are displayed in Figure 1315 In these modelsthe QP order can be introduced in

(i) the sequence of atomic potentials 119881119899 (Figure 13(b))

this arrangement is referred to as the on-site (ordiagonal) model and all the transfer integrals areassumed to be equal over the lattice (119905

119899119899+1= 119905 forall119899)

so that (9) reduces to

120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)

where the common transfer integral value defines thesystem energy scale (ie 119905 equiv 1)

(ii) the sequence of bonds along the chain 119905119899119899+1

(Figures 13(c) and 13(d)) these arrangements arereferred to as the induced (Figure 13(c)) and the stan-dard (Figure 13(d)) transfer (or off-diagonal) modelsrespectively In both cases all the atomic potentials areassumed to be equal (119881

119899= 119881 forall119899) and (9) reduces to

119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9

Figure 12 Self-similarity in the energy spectrum of a InAsGaSbFibonacci superlattice The left panel shows the whole spectrum foran order 119899 = 3 superlattice whereas the central and right panelsshow a detail of the spectrum for superlattices of orders 119899 = 6 and119899 = 9 respectively (from [47] with permission from IOP PublishingLtd)

where without loss of generality the origin of energyis set by the atomic potentials (ie 119881 equiv 0)

(iii) both the atoms and bonds sequences this arrange-ment corresponds to the general case and it is gener-ally assumed that the QP sequence of bonds is deter-mined by the QP sequence of atoms since chemicalbonds between different atoms will depend on theirchemical nature (Figure 13(a)) As a consequence thearrangement of transfer integrals 119905

119860119861= 119905119861119860

and 119905119860119860

issynchronized with 119881

119899 in such a way that the 119905

119899119899+1

sequence is still aperiodic but it does not obey thesame QP sequence that determines the 119881

119899 sequence

(cf Figures 13(c) and 13(d))

Due tomathematical simplicity reasonsmost early worksin the period 1985ndash1990 focused on the two particularversions of the Schrodinger equation given by the on-site andthe standard transfermodels InTable 3we list the parametersgenerally used in the study of different binary QP models

Making use of the trivial relation 120595119899= 120595119899 (9) can be cast

in the convenient matrix form

(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)

where M119899is referred to as the local transfer matrix Thus

for a given system size 119873 the wave function amplitudes can


VA VAVAVB VB VA VA VAVB VB VA VA VBmiddot middot middot

tAB tAAtAB tAB tAB tAB tAB tAA tAB tAB tAA tAB

(a)


t

(b)

V

middot middot middot


(c)

middot middot middottBtA tA tA tB tB tAtA tA tB tA tA tB

V = 0

(d)

Figure 13 Tight-binding Fibonacci chainmodels describing the electron dynamics in terms of on-site energies119881119899and transfer integrals 119905

119899119899plusmn1

in the (a) general case (b) on-site (diagonal) model (c) induced transfer model and (d) standard transfer (off-diagonal) model

Table 3 Atomic potential and transfer integral parameter valuesadopted in different models

Model Potential Transfer integralGeneral 119881

119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1

On site 119881119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1

Standard transfer 119881119860= 119881119861equiv 0 119905

119860= 1 119905

119861= 1

Induced transfer 119881119860= 119881119861equiv 0 119905

119860119860= 119905119860119861

equiv 1

be recursively obtained by successively multiplying the initialvalues 120595

1and 120595

0 according to the expression

(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)

where M119873(119864) is the so-called global transfer matrix There-

fore to solve the Schrodinger equation is completely equiva-lent to calculating products of transfer matrices and the QPorder of the system is naturally encoded in the particularorder of multiplication of these transfer matrices to giveM119873(119864) In this way the noncommutative character of the

matrix product endows the global transfer matrix with afundamental role on the description of QP order effects inthe transport properties of QPS

Indeed within the transfer matrix framework the com-plexity of a given QPS can be measured by the number ofdifferent kinds of local transfer matrices which are necessaryto fully describe it as well as by the particular order ofappearance of these matrices along the chain For instancein the case of the Fibonacci sequence one has two differentlocal transfer matrices in the on-site model (see Figure 13(b))

A equiv (119864 minus 120598 minus1

1 0) B equiv (

119864 + 120598 minus1

1 0) (25)

where without loss of generality the origin of energies isdefined in such a way that 119881

119860= 120598 = minus119881

119861(Table 3) The

number of required local transfer matrices increases to three

in the transfer models Thus for the standard model we have(119905119860equiv 1)

C equiv (119864 minus1

1 0) D equiv (

119864 minus119905119861

1 0)

E equiv (119864119905minus1

119861minus119905minus1

119861

1 0

)

(26)

whereas for the induced transfer model one gets

C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860

119905119860119861 and the energy scale is given by 119905

119860119861equiv 1

Finally the number of local transfer matrices rises to four inthe general case namely

A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1

(119864 minus 120598) minus120574minus1

1 0

)

(28)

The presence of the matrices A and B in the general modelindicates that the on-site model can be naturally obtainedas a particular case corresponding to the condition 120574 = 1which reduces Z = Y = A In a similar way the inducedtransfer model can be obtained from the general one byimposing the condition 120598 = 0 which reduces Z = KY = L and A = B = C On the contrary the standardtransfermodel cannot be straightforwardly obtained from thegeneral one because the transfer integrals sequence 119905

119899119899+1

corresponding to the general case does not coincide with theatomic potential sequence 119881

119899 (Figure 13(c)) Accordingly

the general on-site and induced transfer models can beregarded as sharing the same lattice topology whereas thestandard transfer model does not We also note that thematricesA B andC are unimodular (ie their determinantsequal unity) while the remaining local transfer matrices arenot

In the case of the on-site model the transfer matrixformalism allows one to establish a one-to-one correspon-dence between the atomic potentials sequence and the local


Table 4 The first Fibonacci numbers from 119895 = 1 to 119895 = 24

1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025

transfer matrices sequence (see Figure 13(b)) so that theglobal transfer matrix reads

M119873 (119864) =

1

prod

119899=119873

M119899= sdot sdot sdotBAABAABABAABA (29)

Since A and B are both unimodular it can be stated that theglobal transfer matrix for the on-site model belongs to the119878119897(2R) group It is ready to check that the matrices string in(29) can be recursively obtained by concatenation accordingto the expression M

119899+1= M119899minus1

M119899 starting with M

0= B

and M1

= A so that if 119873 = 119865119895 where 119865

119895is a Fibonacci

number obtained from the recursive law119865119895= 119865119895minus1

+119865119895minus2

with1198651= 1 and 119865

0= 1 the number of A matrices is 119899

119860= 119865119895minus1

and the number of B matrices is 119899119861= 119865119895minus2

For the sake ofinformation in Table 4 we list the first Fibonacci numbers

Making use of (26) the global transfer matrix for thestandard transfer model reads (see Figure 13(d))

M119873 (119864) =

1

prod

119899=119873

M119899= sdot sdot sdotCDEDECDE equiv sdot sdot sdot BAABA

(30)

where we have defined B equiv C and introduced the newmatrix[48]

A equiv DE = 119905minus1

119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)

which is unimodular For119873 = 119865119895there are 119899

119860= 119865119895minus2

matricesof type A and 119899

119861= 119865119895minus3

matrices of type B Therefore byintroducing the matrices A and B we are able to express theglobal transfer matrix corresponding to the standard transfermodel in terms of just two different unimodular matricesarranged according to the Fibonacci sequence as we didfor the on-site model Therefore M

119873(119864) also belongs to the

119878119897(2R) group in this caseIn a similar way making use of (27) we can express the

global transfer matrix corresponding to the induced transfermodel as

M119873 (119864) =

1

prod

119899=119873

M119899= sdot sdot sdotCCKLCKLCCCKLC

equiv sdot sdot sdot S119861S119860S119860S119861S119860

(32)

wherewe have introduced the auxiliary unimodularmatrices

S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)

so thatM119873(119864) belongs to the 119878119897(2R) group for the induced

transfer model as well In this case for119873 = 119865119895we have 119899

119860=

119865119895minus3

matrices S119860and 119899119861= 119865119895minus4

matrices S119861

Finally making use of (28) we can translate the potentialssequence 119881

119860 119881119861 119881119860 119881119860 119881119861 describing the atomic order

of the general Fibonacci lattice to the local transfer matricesproduct sdot sdot sdotABZYBZYBABZYB describing the behavior ofelectrons moving through it (see Figure 13(a)) In spite of itsgreater apparent complexity we realize that by renormalizingthis transfer matrix sequence according to the blockingscheme R

119860equiv ZYB and R

119861equiv AB we get the considerably

simplified global transfer matrix

M119873 (119864) =

1

prod

119899=119873

M119899= sdot sdot sdotR

119861R119860R119860R119861R119860 (34)

Thus we can express the M119873(119864) matrix of the Fibonacci

general model in terms of just two matrices instead of theoriginal four [49]The subscripts in theR] matrices are intro-duced to emphasize the fact that the renormalized transfermatrix sequence is also arranged according to the Fibonaccisequence and consequently the topological order presentin the original lattice is preserved by the renormalizationprocess In fact if 119873 = 119865

119895is the number of original lattice

sites it can then be shown by induction that the renormalizedmatrix sequence contains 119899

119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4

matrices R119861 Quite remarkably the renormalized

matrices

R119861= (

1198642minus 1205982minus 1 120598 minus 119864

120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)

2minus 1205742] + 120598 minus 119864 are both

unimodular so that M119873(119864) belongs to the 119878119897(2R) group in

the general Fibonacci model as wellAs we see the matrix elements of the different renormal-

ized matrices are polynomials of the electron energy so thatone may expect that these matrices could adopt particularlysimple forms for certain energy values To explore such apossibility in a systematic way it is convenient to explicitly


evaluate the commutators corresponding to the differentFibonacci lattices shown in Figure 13 [49 50]

[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)

Thus we see that both the on-site and the standard transfermodels are intrinsically noncommutative for their commuta-tors only vanish in the trivial 119881

119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1

cases respectively On the contrary in the induced transfermodel the S] matrices commute for the energy value 119864 = 016In that case

S119861 (0) = (

minus1 0

0 minus1) = minusI S

119860 (0) = (

0 120574

minus120574minus1

0

) (37)

and (34) can be expressed as

M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)

120574minus1119880119899119860minus1 (0) minus119880

119899119860minus2 (0)

)

(38)

where we have made use of the Cayley-Hamilton theorem17Equation (38) guarantees that the 119864 = 0 state belongs tothe energy spectrum since | trM

119873(0)| = |2119880

119899119860minus2(0)| =

|2 sin((119899119860minus 1)1205872)| le 2 forall119899

119860 On the other hand making

use of the relationships 1198802119899(0) = (minus1)

119899 and 1198802119899+1

(0) = 0 werealize that (38) can take on two different forms dependingon the parity of the 119899

119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)

Thus the wave function has a simple form and takes onvalues plusmn1 or plusmn120574plusmn1 (assuming 120595

0= 1205951= 1) The 119860 sites have

|120595119899|2= 1 or 120574minus2 whereas the 119861 sites have |120595

119899|2= 1 or 1205742

[50] Accordingly this is an extended state which propagatesthrough the Fibonacci induced transfer model

Finally according to expression (36) there exists alwaysone energy satisfying the relation

119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)

for any realization of the general model (ie for any com-bination of 120598 = 0 and 120574 = 1 values) For these energies thecondition [R

119860R119861] = 0 is fulfilled and making use of the

Cayley-Hamilton theorem the global transfer matrix of thesystemM

119873(119864lowast) equiv R119899119860119860R119899119861119861 can be explicitly evaluated From

the knowledge of 119872119873(119864lowast) the condition for the considered

energy value to be in the spectrum | tr[M119873(119864lowast)]| le 2 can

be readily checked We will discuss in detail the properties ofthese states in Sections 33 and 42

22 Fractal Lattices Prior to the discovery of QCs it wassuggested by some authors that fractal structures whichinstead of the standard translation symmetry exhibit scaleinvariance may be suitable candidates to bridge the gapbetween crystalline and disordered materials [51] Such apossibility was further elaborated in subsequent works oninhomogeneous fractal glasses [52 53] a class of structureswhich are characterized by a scaling distribution of poresizes and a great variety in the site environments From thisperspective it is interesting to compare the physical propertiesrelated to these two novel representatives of the orderings ofmatter namely QCs and fractals

Thus both numerical and analytical evidences of local-ized critical and extended wave functions alternating ina complicated way have been reported for several fractalmodels [54ndash62] In addition it was reported that the interplaybetween the local symmetry and the self-similar nature of afractal gives rise to the existence of persistent superlocalizedmodes in the frequency spectrum [63] This class of statesarise as a consequence of the fact that the minimum pathbetween two points on a fractal does not always follow astraight line [64] Consequently the general question regard-ing whether the nature of the states might be controlled bythe fractality of the substrate is an interesting open questionwell deserving further scrutiny

Let us start by considering the celebrated triadic Cantormodel which can be obtained from the substitution rule119860 rarr 119860119861119860 and 119861 rarr 119861119861119861 leading to the sequences

119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860

997888rarr sdot sdot sdot

(41)

The number of letters contained in the 119896 order fractalgeneration is 119873 = 3

119896 By inspecting these strings we realizethat these sequences differ from the binary QP sequencesconsidered in Section 22 in the sense that as the systemgrows on the subclusters of 119861s grow in size to span theentire 1D space Therefore in the thermodynamic limit thelattice may be looked upon as an infinite string of 119861 atomspunctuated by 119860 atoms which play the role of impuritieslocated at specific sites determined by the Cantor sequence[60 65]

The Schrodinger equation corresponding to the on-siteversion of the triadic Cantor lattice can be expressed in termsof a global transfer matrix composed of the two local transfermatrices A and B given by (25) as follows

M3 (119864) = ABA

M9 (119864) = ABAB3ABA

M27 (119864) = ABAB3ABAB9ABAB3ABA

(42)

One then sees that for those energy values satisfying thecondition B3 = plusmnI these global transfer matrices reduce to


12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9

Figure 14 Tight-binding homogeneous Vicsek fractal modeldescribing the electron dynamics in terms of on-site energies 120598

0

and transfer integrals 119905 The fractal generation order is given by theinteger 119896 while119873 stands for the number of atoms in the decimatedchains

that corresponding to a periodic structure with a unit cell11986011986111986018 Accordingly if the energies satisfying the conditionB3 = plusmnI also belong to the spectrum of the 119860119861119860 periodiclattice (ie they satisfy the property | tr(ABA)| le 2) thenthese energies will correspond to extended states Makinguse of the Cayley-Hamilton theorem the above resonancecondition can be expressed in terms of the matrix equation

B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =

(119864 + 120598)119905minus12 By solving (43) one obtains 119909 = minus12 when

B3 = I and 119909 = 12 when B3 = minusI In a similar wayone can get other extended states by setting B3

119896

= plusmnIwith 119896 gt 1 and solving the resulting polynomial equationThus the on-site triadic Cantor set admits a countable setof extended states whose number progressively increases asthe system size increases This result properly illustrates thatthe scale invariance symmetry related to self-similar topologycharacteristic of a Cantor lattice favours the presence ofextended states propagating through the structure

An illustrative fractal lattice extending in more than onedimension is provided by the so-called Vicsek lattice Thepattern corresponding to its two first generations is shownin Figure 14 where we see that its basic building block isformed by a cross composed of five identical atoms (on siteenergy 120598

0) coupled to the central one with identical transfer

integrals 119905 Quite remarkably this fractal lattice can be exactlymapped into a 1D chain of atoms by systematically decimatingthe upper and lower branches around the central site atany fractal generation as it is illustrated on the right handof Figure 14 for first- and second-generation lattices Theresulting effective 1D decimated chain is composed of a setof 119873 = 3

119896 atoms with different on-site energy values whichare coupled to each other by identical transfer integrals 119905

The resulting decimated lattice exhibits a hierarchical struc-ture so that the successive values of the on-site energy series1205980 1205981 1205982 arrange themselves in a self-similar pattern

where 120576119896arises out of the decimation of bigger and bigger

clusters of atoms around the central sites at each 119896 generationorder of the fractal and are given by the fraction series

1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052

119864 minus 1205981minus 1199052 (119864 minus 120598

1minus 1199052 (119864 minus 120598

0))

(44)

By inspecting Figure 14 we realize that by imposing thecondition 120598

1= 1205980the decimated lattice corresponding to

the first generation Vicsek fractal reduces to a monatomicperiodic chain whereas by imposing the condition 120598

2= 1205981

the decimated lattice corresponding to the second generationVicsek fractal reduces to a binary periodic chain the unit cellof which is precisely given by the 119896 = 1 decimated chainAccordingly the eigenstates corresponding to these reso-nance energies will be extended states in their correspondingeffective periodic chains Obviously the same procedure canbe extended to higher order generations to reduce the originalhierarchical lattice to a periodic lattice containing as manydifferent types of atoms as the generation order In thisregard one may consider that the topological complexity ofthe original Vicsek fractal is properly translated to a higherchemical diversity as a consequence of the renormalizationprocesses Therefore as the Vicsek fractal grows larger wecan obtain a numerable infinite set of extended states in thethermodynamic limit The energies corresponding to thesestates can be obtained from (44) by imposing the condition120598119896+1

= 120598119896 119896 ge 1 iterativelyThe same condition can be arrived

at by considering the local transfermatrices corresponding tothe on-site hierarchical lattice model which have the form

V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)

and calculating the commutator

[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)

which only commutes when 120598119896+1

= 120598119896 Since the global

transfer matrix of the decimated 1D chain is given as aproduct of local transfer matrices of the form

M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V

0V1V0V0V2V0V0V1V0V0V1V0V0V3

times V0V0V1V0V0V1V0V0V2V0V0V1V0

M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)

the hierarchical distribution of the renormalized on-site ener-gies in the decimated lattice guarantees that this hierarchicalstructure is inherited by the corresponding extended statessatisfying the commutation condition stated above

As a final model example we will consider the tight-binding model on the Koch lattice introduced by Andradeand Schellnhuber [66] The motivation for this choice stemsfrom the fact that this model Hamiltonian can also beexactly mapped onto a linear chain and the correspondingelectron dynamics expressed in terms of just two kinds ofrenormalized transfer matrices In this way we can use thesame algebraic approach discussed in the study of electrondynamics in general Fibonacci systems in the previoussection The model is sketched in Figure 15 and its tight-binding Hamiltonian is given by [66]

119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)

times [|119899 minus 1⟩⟨119899 + 1| + |119899 + 1⟩ ⟨119899 minus 1|]

(48)

where 120582 is the cross-transfer integral introduced by Gefen etal [67] (indicated by dashed lines in Figure 15(a)) and

119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)

with 119896 ge 2 and minus41198962 le 119899 le 4

1198962 describing the

effective next-nearest-neighbor interaction in the 119896th stageof the fractal growth process The main effect of allowingelectron hopping across the folded lattice is the existence ofsites with different coordination numbers along the latticea characteristic feature of fractals which is not shared byQP lattices Depending on the value of their respectivecoordination numbers we can distinguish twofold (circles)threefold (full triangles) and fourfold (squares) sitesWe thennotice that even sites are always twofold a fact which allowsus to renormalize the original latticemapping it into the linearform sketched in Figure 15(b) [66] The transfer integralsrepresented by single bonds appear always isolated from oneanother The transfer integrals represented by double bondscan appear either isolated or forming trimers Consequentlythere are three possible site environments in the renormalizedKoch latticewhich in turn define three possible types of localtransfer matrices labelled F G and H in Figure 15(b) Nowby introducing the matrices P equiv HG and PQ equiv FF it canbe shown by induction that the global transfer matrix at anygiven arbitrary stage 119896 of the fractal growth processM

119896 can

be iteratively related to that corresponding to the previousstageM

119896minus1 by the expression [66]

Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)

with 119896 ge 2 and M1

equiv P In this way the topologicalorder of the lattice is translated to the transfer matricessequence describing the electron dynamics in a natural wayThematrices P andQ are unimodular for any choice of 120582 andfor any value of the electron energy 119864 and their commutatorreads [55]

[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033

times ((2 minus 119864

2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)

where 119903 equiv 1+120582119864 andwe have defined the origin of energies insuch a way that the transfer integrals along the original chainequal unity By comparing (51) with the commutator [R

119860R119861]

for the general Fibonacci lattice given by (36) we see that theKoch fractal lattice has a larger number of resonance energiesnamely 119864 = 0 119864 = plusmnradic2 and 119864 = minus2120582 as compared to thesingle 119864

lowastvalue obtained for the Fibonacci one

23 Quasiperiodic Optical Devices In order to fully appreci-ate the fingerprints of long-range aperiodic order the studyof classical waves propagating through an aperiodic substrateoffers a number of advantages over the study of quantumelementary excitations

Consequently light transmission through aperiodicmedia has deserved an ever-increasing attention in orderto understand the interplay between optical propertiesand the underlying aperiodic order of the substrate[18 68ndash71] To this end the mathematical analogy betweenSchrodinger equation (5) andHelmholtz equation describinga monochromatic electromagnetic wave of frequency 120596

propagating in a lossless dispersionless medium with avariable refractive index profile 119899(119909)

1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)

where E is the transversal component of the electric field119896is the wave vector in the XY plane (perpendicular to

the propagation direction 119911) and 119888 is the vacuum speed oflight provides a powerful tool to relate previous knowledgeabout electron motion in superlattices to electromagneticwaves propagating in multilayers (Figure 16) We note thatthe refractive index of the different layers is the physicalmagnitude relating the aperiodic sequence describing thestacking order along the multilayer (Figure 16(a)) and theresulting aperiodic function describing themultilayerrsquos opticalprofile (Figure 16(b)) Thus the isomorphism of Shrodingerand Helmholtz equations provides a helpful analogy involv-ing basic concepts in modern optoelectronics [72]

3 Nature of States in Aperiodic Systems

Thenotion of critical wave function has evolved continuouslysince its introduction in the study of QP systems (seeSection 12) leading to a somewhat confusing situation Forinstance references to self-similar chaotic lattice-like or


middot middot middotmiddot middot middot120582

minus2minus1

0

12

4

6

8

h

(a)

middot middot middotmiddot middot middot

G H G H G H G HF F

minus7 minus1 1 7

(b)

Figure 15 (a) Sketch of the Koch latticemodel consideredThe different sites are labeled by integers (b) Sketch of the renormalization schememapping the Koch lattice into a linear chain (from [55] with permission fromMacia copy 1998 by the American Physical Society)

B A A A A A A AB B B B B A A A AB BAA

Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)

Figure 16 (a) Sketch of a Fibonacci dielectric multilayer grown along the 119911 direction (b) refractive index profile 119899(119911) for an electromagneticwave propagating through the structure and (c) electronic potential profile119881(119911) for an electron propagation along the growth direction (from[68] Macia with permission from IOP Publishing Ltd)

quasilocalized wave functions can be found in the literaturedepending on the different criteria adopted to characterizethem [48 73ndash75] Generally speaking critical states exhibita rather involved oscillatory behavior displaying strong spa-tial fluctuations which show distinctive self-similar features(Figure 5) Thus the wave function is peaked on short chainsequences but peaks reappear far away on chain sequencesshowing the same lattice ordering As a consequence thenotion of an envelope function which has been the mostfruitful in the study of both extended and localized states ismathematically ill defined in the case of critical states andother approaches are required to properly describe them andto understand their structure

On the other hand as we have seen in Section 14 froma rigorous mathematical point of view the nature of a stateis uniquely determined by the measure of the spectrum towhich it belongs In this way since it has been proven thata number of QPS have purely singular continuous energyspectra we must conclude that the associated electronicstates cannot be strictly speaking extended in Blochrsquos senseHowever this fact does not necessarily imply that statesbelonging to singular continuous spectra of differentQPSwillbehave in exactly the same way from a physical viewpoint

In fact electronic states can be properly classified accord-ing to their related transport properties Thus conduct-ing states in crystalline systems are described by periodicBloch functions whereas insulating systems exhibit exponen-tially decaying functions corresponding to localized statesWithin this scheme the notion of critical states is somewhat

imprecise because critical states exhibit strong spatial fluctu-ations at different scales In this regard a first step towardsa better understanding of critical states was provided by thedemonstration that the amplitudes of critical states in on-site Fibonacci lattices do not tend to zero at infinity but arebounded below through the system [28] This result suggeststhat the physical behavior of critical states is more similar tothat corresponding to extended states than to localized onesIndeed the possible existence of extended states in severalkinds of aperiodic systems has been discussed in the last yearsspurring the interest in the precise nature of critical wavefunctions and their role in the physics of aperiodic systems[49 50 55 73 76ndash79] As a result arguments supporting theconvenience of widening the very notion of extended statein aperiodic systems to include critical states which are notBloch functions have been put forward [49 55]

31 Eigenstates in On-Site Fractal Lattices How do the eigen-functions of self-similar fractal lattices compare with thosereported for the Aubry-Andre model at the critical point InFigure 17(a) we plot the wave function amplitudes for oneof the solutions of (43) for a lattice with 119873 = 3

5= 243

lattice sites The distribution shows an interesting patternwhich mimics the topological atomic arrangement of thelattice itself in the sense that all peaks group close togetherin clusters containing 1 3 or 27 peaks which are also powersof three like what occurs for the 119861s strings interspersed inthe lattice For this reason these types of states are sometimesreferred to as lattice-like in the literature19 When comparing


0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)

Figure 17 (a) A lattice-like distribution of amplitudes for the extended state 119864 = 0 in the on-site model of a119873 = 243 triadic Cantor lattice(b) An extended wave function corresponding to the energy 119864 = minus13473 obtained by setting B9 = minusI In both cases 119881

119860= 0 119881

119861= minus1 and

119905 = 1 (adapted from [60] courtesy of Arunava Chakrabarti with permission from Elsevier)

the wave function distribution amplitudes of this state forlattices of different lengths it was reported that the lattice-like feature recurrently appears and disappears dependingon the generation order of the fractal lattice being presentfor those lattices whose length is given by the series 119873 =

32119896+1 with 119896 ge 1 [60] For other energy values one obtainsextended states which however do not generally exhibit thelattice-like property as it is shown in Figure 17(b) Thereforethe lattice-like property is by no means a generic propertyof extended states in triadic Cantor lattices In addition bycomparing Figures 5 and 17 we see that the wave functionsreported for Cantor lattices lack the characteristic self-similararrangements of peaks observed in the critical states of theAubry-Andre model

32 Self-SimilarWave Functions in Fibonacci Transfer ModelsThe wave function distribution amplitudes for the statelocated at the band centre (119864 = 0) in the Fibonacci standardtransfermodel (see Figure 13(d)) were earlier studied in detailby Kohmoto and coworkers and it is shown in Figure 18One can readily appreciate a series of main peaks which aresequentially found as 119899 is increased The peak values aregiven by powers of the parameter 119905

119861119905119860measuring the ratio

between the long and short bonds (in Figure 18 119905119861

= 2 sothat the peaks take on 2

119898 values) In order to analyze thespatial distribution of the wave function amplitudes we willsplit the lattice sites into two complementary sets dependingon whether 119899 is (or not) a Fibonacci number Due to theself-similar structure of Fibonacci lattices the global transfermatrix for lattice sites corresponding to Fibonacci numberscan be iteratively obtained from the relation [80]

M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)

On the other hand making use of the property that anyinteger can be expressed as a sum of Fibonacci numbers inthe form 119899 = 119865

1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894

with 1198971gt 1198972gt sdot sdot sdot gt 119897

119894isin N

the global transfer matrix for a non-Fibonacci 119899 lattice site isgiven by

M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)

Let us first consider the lattice sites corresponding toFibonacci numbers Since we are considering the 119864 = 0 state(31) reads

M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)

Then making use of (53) one gets

M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)

accordingly M55(0) = M

21(0)M

34(0) = M

1(0)M

2(0) =

M3(0) and the entire sequence of global transfer matrices

repeats once again Therefore we get the six-cycle propertyM119865119895+6

(0) = M119865119895

(0) relating the global transfer matrices of


1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)

Figure 18 Self-similar features in thewave function located at the center of the spectrum119864 = 0 in the Fibonacci standard transfermodel with119905119861= 2 In (a) the wave function for a lattice composed of 119865

21= 17711 sites is plotted A portion of this wave function comprised within the

sites indicated by the arrows around the highest peak located at 119899 = 5472 is shown in (b)This amplitude distribution is identically reproducedat progressively smaller scales in (c) and (d) after a proper rescaling (and renumbering of the lattice sites) (from [48] with permission fromKohmoto et al copy 1987 by the American Physical Society)

progressively longer sections along the Fibonacci chain Thesize119873 = 55 = 119865

9thus represents the minimum length neces-

sary for the properties stemming from thismatrices repeatingpattern tomanifest themselvesThe structure of (56)matricesalso guarantees that the wave function amplitudes can onlytake on the values plusmn1 or plusmn119905plusmn1

119861times 120595

0or 1205951at these lattice

sitesThe situation is different for lattice sites which do not

correspond to Fibonacci numbers In fact there are aseries of peaks located at the sites given by the formula

119899lowast= 1198653+ 1198656+ sdot sdot sdot + 119865

3119901 119901 = 1 2 3 whose corresponding

global transfer matrices (for 119901 gt 2) read

M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)

where we have made use of (54) and the six-cycle propertyM119865119895+6

(0) = M119865119895

(0) According to (24) at these lattice sitesthe wave function amplitudes obey the geometric series120595

119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1

1198611205951 These two series of lattice sites

give the fundamental structure of the wave function as theydetermine the series of progressively higher peaks which areencountered sequentially as 119899 is increased [80] Indeed as oneconsiders longer and longer lattices one obtains higher andhigher peaks as it is seen in Figure 18(a)This implies that thewave function is not bounded above in the thermodynamiclimit119873 rarr infin

A typical self-similar structure which can be described bymeans of a proper scale transformation of the wave functionamplitudes can be identified around the highest peaks as itis illustrated in Figures 18(b)ndash18(d)20 The self-similarity ofthis wave function can be described precisely as follows if oneassumes 120595

0= 1205951= 1 then the modulus of the wave function

is invariant under the scale transformation [80]

119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)

where 120591 = (1 + radic5)2 is the golden mean Accordingly thewave function amplitude obeys a power-law distribution ofthe form 120595

119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860

= 1 one gets 120585 = 0480) [80] which defines a localenvelope of the wave function around the sites with high peakvalues A similar critical exponent for the envelope of thewave functionwas obtained by introducing a renormalizationgroupmethod to study the Fibonacci standard transfermodel[81 82] The reason for a power-law behavior is basically thatthe global transfer matrices of the form

(119905minus2119898

1198610

0 1199052119898

119861

) (59)

corresponding to lattice sites 119899lowast= sum119865

3119901containing an even

number of Fibonacci numbers in the sum (eg 119899lowast= 16304

and 5472 in (57)) have an eigenvalue which is greater than

1 17711

n

|120595n|

Figure 19 Example of a non-self-similar wave function correspond-ing to 119864 = 0267958 in the Fibonacci standard transfer model with119905119861= 2 (adapted from [48] with permission from Kohmoto et al copy

1987 by the American Physical Society)

unity21 Therefore although the wave function does not growand only takes the values plusmn1 plusmn(119905

119861119905119860)plusmn1 at Fibonacci sites

it grows as a power-law at non-Fibonacci sitesAlthough these exact results have been obtained for the

special energy 119864 = 0 there are many other energies at whichthe wave functions behave similarlyThose energies are at thecenters of the main subbands in the energy spectrum whichprogressively appear as the system grows longer Self-similarwave functions were also reported at the energy spectrumedges where the global transfer matrices exhibit a two-cyclerecurrence [48]

However as far as 1986 the notion of critical wavefunction in Fibonacci systems was not yet clearly defined asit is illustrated by this excerpt by Kohmoto [42]

There are two types of unusual wave functionsself-similar and chaotic The self-similarity existsfor wave functions corresponding to the centerand the edge of each subcluster of the spectrumThe wave function at each center is similar to theexact wave function at 119864 = 0 In the same wayall the wave functions at the edges are similar toeach other Chaotic states exist almost everywherein the spectrum They are the remaining stateswhich are not at the centers or the edges of everysubband in the spectrum These states are notnormalizable and do not have apparent scalingpropertiesThe nature of these chaotic states is notwell understood yet

An example of a non-self-similar wave function charac-terized by strong spatial fluctuations of the wave functionamplitudes is shown in Figure 19 The overall long-range


structure of this wave function can be split into two mainregions including 119865

20= 10946 and 119865

19= 6765 lattice sites

each which are respectively located on the left (right) ofthe arrow shown in the figure The wave function amplitudescorresponding to these two regions exhibit distinctive mirrorsymmetry features with respect to the dashed vertical linesAlthough one cannot see any evidence of self-similar patternsin this amplitudes distribution by simple visual inspectionthere exist more powerful methods to disclose short-rangeinternal correlations in such a distribution Two wide proce-dures used to this end are based on the so-called multifractaland wavelets analysis respectively (see Section 15)

The use of multifractal methods to analyze the electronicstates in Fibonacci lattices provided conclusive evidence onthe existence of different kinds of critical states dependingon their location in the highly fragmented energy spectraThus while states located at the edges or the band centersof the main subbands exhibit a distinctive self-similar spatialstructure most of the remaining states do not show anyspecific pattern [48]

33 Extended States in General Fibonacci Lattices InSection 21 we saw that whereas both the on-site andstandard transfer models are intrinsically noncommutativeone could always find the special energy value given by (40)satisfying [R

119860R119861] = 0 in general Fibonacci lattices In this

case the renormalized matrices given by (35) adopt the form

R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)

Since 119902 = 0 in the general Fibonacci lattice (the choice 120598 =

0 leads to the induced transfer model) R119861cannot adopt a

diagonal form but R119860becomes diagonal for 119902 = plusmn1 In that

case R119860= plusmnI and the global transfer matrix reads

Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)

where we have made use of the Cayley-Hamilton theorem toevaluate the required power matrix

Quite interestingly the matrices given by (60) can beexpressed as powers of a commonmatrix themselves namelyR119861= R2 and R

119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)

is a unimodular matrix [83] In this way making use ofthe Cayley-Hamilton theorem the global transfer matricescorresponding to the energies given by (40) can be readilyexpressed in the closed form

M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1

minus119880119873minus2

) (64)

where 119880119896(119909) are Chebyshev polynomials of the second kind

and

119909 =119902

2=

1

2radic1198642lowastminus 1205982 equiv cos120601 (65)

Taking into account the relationship 119880119899minus 119880119899minus2

= 2119879119899

between Chebyshev polynomials of the first and secondkinds from (64) one gets tr[M

119873(119864lowast)] = 2 cos(119873120601)

Consequently we can ensure that energies determined by(40) belong to the spectrum in the QP limit (119873 rarr infin)

The charge distribution shown in Figure 20(a) corre-sponds to a state of energy 119864lowast

1= minus125 in a general Fibonacci

chain with119873 = 11986516

= 1 597 sites and lattice parameters 120574 = 2

and 120598 = 075 Figure 20(b) shows the charge distributioncorresponding to a state of energy 119864

lowast

2= minus56 in a general

Fibonacci lattice of the same length and model parameters120574 = 2 and 120598 = 05 The overall behavior of the wave functionsamplitudes clearly indicates their extended character Indeedthe presence of identity matrices in the product defining theglobal transfermatrix favours the existence of extended statesspreading relatively homogeneously throughout the latticeAt this point it is worthwhile to mention that despite itsappearance this wave function is nonperiodic the sequenceof values taken by the wave function amplitude is arrangedaccording to a QP sequence

An illustrative example of the energy spectra of generalFibonacci chains for different values of the 120574model parameteris shown in Figure 21 Its characteristic fragmentation schemeis clearly visible One appreciates that the extended statesgiven by (40) are located in the densest regions of the phasediagram The corresponding band edge states are obtainedfrom the condition | tr[M

119873(119864lowast)]| = 2 and for an arbitrary

119873 value they are given by the condition 120601 = 0 120587 rArr

119909 = plusmn1 This condition is satisfied by the model parameterscombinations 120598120574 = plusmn(1 minus 120574

2) respectively For the argument

values 119909 = plusmn1 the Chebyshev polynomials considerablysimplify to the form119880

119899(1) = 119899+1 and119880

119899(minus1) = (minus1)

119899(119899+1)

and the global transfer matrices can be expressed as

M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873

minus120574minus1119873 1 minus 119873

)

(66)

where 119864+

lowast(119864minuslowast) denote the eigenstate for the top (bottom)

band edge respectively In both cases the wave functionamplitudes grow linearly with the lattice site


0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)

Figure 20 Electronic charge distribution in general Fibonacci lattices with 119873 = 11986516and (a) 120574 = 2 120598 = 075 and 119864

lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast

2= minus56 The wave function amplitudes have been calculated with the aid of the matrix formalism making use of the initial

conditions 1205930= 0 and 120593

1= 1 (from [49] with permission fromMacia and Domınguez-Adame copy 1996 by the American Physical Society)

minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3

Figure 21 Phase diagram (120574 in abscissas and 119864 in ordinates) for ageneral Fibonacci chain with 119873 = 34 and 120598 = 05 The energiescorresponding to transparent wave functions are marked with athin white line (courtesy of Roland Ketzmerick from [46] withpermission from CRC Taylor amp Francis group)

4 The Role of Critical States inTransport Properties

41 Signatures of Quasiperiodicity As we saw in Section 12the nature of a wave function was at first established byattending to its spatial structure This criterion has proveduseful as far as it was possible to assign a precise relationshipbetween such a spatial structure and its related transportproperties Thus the amplitudes distribution of a Bloch

extended state fits a periodically modulated pattern whereasfor a localized state this distribution decays exponentiallyaccording to a well-defined localization length Thereforefor both localized and extended states the correspondingwave functions admit a global description in terms of anenvelope function which is characterized by a specific spatialparameter such as the localization length or the periodrespectively On the contrary a global description of thewave function in terms of an envelope function is no longerpossible in general for the eigenstates belonging to aperiodicsystems due to the fluctuating structure shown by theiramplitude distributions

As we have previously mentioned from a physical view-point electronic states can be more appropriately classifiedaccording to their transport properties which are of courseclosely related to the spatial distribution of the wave functionamplitudes as well One can then reasonably expect that therich structure displayed bymost eigenstates in QPS should bereflected to some extent in their related transport propertiesIn order to explore this issue Sutherland and Kohmotostudied the electrical resistance of the six-cycle and two-cycle states located at the center and at the band-edges of theallowed subbands in a binary Fibonacci alloy [84] To thisend they made use of the dimensionless Landauer formula

120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)

where 119872119894119895(119864) are the elements of the global transfer matrix

They found that the Landauer resistance is bounded aboveby a quantity which scales according to a power-law as afunction of the length of the system 120588

119873(119864) le 120588

0119873120585 with


120585 = ln 119905 ln 120591119876 where 120591 is the golden mean and 119876 = 6

(119876 = 2) for six-cycle (two-cycle) eigenstates On the basisof these exact results valid for a subset of eigenstates it wasconjectured that the resistance as a function of the chain sizewas bonded by a power of the sample length for all statesbelonging to the system spectrum

This conjecture was subsequently rigorously confirmedby Iochum and Testard [28]Therefore the power-law scalingof the resistivity can be properly regarded as a fingerprintof critical states in Fibonacci lattices On the other side theplot of the Landauer resistance as a function of the systemsize shows self-similar fluctuations as a consequence of thehighly fragmented structure of their energy spectrum as it isillustrated in Figure 22

Inspired by these results the average transport propertiesin Fibonacci lattices were studied by considering the influ-ence of a number ] of wave functions on the mean value ofthe Landauer resistance for a given system length accordingto the average

⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)

where 119864119894indicates the energy of the corresponding states It

was obtained that the average Landauer resistance also scalesas ⟨120588119873⟩ sim 119873

119886 (119886 gt 0) [85] The power-like increase of theresistance indicates that the electrical conductance vanishesin the limit119873 rarr infin This trend qualitatively agrees with thelarge electrical resistivity values observed inmost icosahedralquasicrystals containing transition metals [46 86]

Another magnitude which has been profusely consideredin the study of QPS is the transmission coefficient given by

119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)

where one assumes the aperiodic chain to be sandwichedbetween two periodic chains (playing the role of metalliccontacts) each one with on-site energy 120576

1015840 and transferintegral 1199051015840 The dispersion relation of the contacts is thengiven by 119864 = 120576

1015840+ 21199051015840 cos 120581 The transmission coefficient

is a useful quantity to describe the transport efficiency inquantum systems Nonetheless 119879

119873(119864) is usually difficult to

be directly measured experimentally though it is directlyrelated to the dimensionless Landauer resistance through theexpression

120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)

Accordingly for those states having a unity transmissioncoefficient 119879

119873(119864) = 1 usually referred to as transparent

states one expects a ballistic transport throughout the latticeCertainly this condition is fulfilled by extended Bloch func-tions describing extended states But what can be said aboutthe possible existence of transparent states in fractal andQPS

42 Transparent Extended States in Aperiodic Systems Let usnow consider a class of wave functions belonging to eitherQPor fractal systems which satisfy the condition 119879

119873(119864) = 1 so

that they can be regarded as extended from a physical point ofview though their wave function amplitudes distribution alsoexhibits characteristic QP features This result prompts oneto properly widen the notion of transparent wave functionto include electronic states which are not Bloch functionsthereby clarifying the precise manner in which the QPstructure of these wave functions influences their transportproperties

Indeed from a detailed numerical study of the triadicCantor lattice it was found that the transmission coefficientbecomes equal to unity for many energy eigenvalues corre-sponding to extended eigenfunctions of the system as it isillustrated in Figure 23 However it should bementioned thatthere are also values of the transmission coefficient whichlie between zero and one For example the transmissioncoefficient corresponding to the energy 119864 = 0 in the on-sitemodel with 119881

119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)

for 119896 gt 1 where the 119896 is the fractal generation orderdetermining the chain length 119873 = 3

119896 Accordingly when119896minus1 is amultiple of 3 the state becomes transparent elsewhereone gets 119879

119873(0) = 45 = 08 as it is the case of the eigenstate

shown in Figure 17 for a lattice with 119873 = 35 sites Therefore

among the extended wave functions belonging to the triadicCantor lattice spectrum energy one can find both transparentand nontransparent states depending upon the adopted 119873

value [60]A similar result is obtained when considering the self-

similar state corresponding to 119864 = 0 in the standard transfermodel of the Fibonacci binary alloy (Figure 18) As we sawin Section 32 its global transfer matrix exhibits a six-cycleproperty so that depending on the 119873 value there exist fivepossible types of resultingM

119873(0) matrices as prescribed by

(56) By plugging the correspondingmatrix elements119872119894119895into

(69) one gets 119879119873lowast(119864) = 1when the transfer matrix adopts the

simple form given byM1(0) whereas one has

119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)

elsewhere Since this expression is less than unity for alltransfer integral values different from the trivial periodic case1199052

119861= 122 one concludes that 119879

119873(0) lt 1 for a large number

of 119873 = 119865119895values In fact the state shown in Figure 18

(119873 = 17711 = 11986521) is not a transparent one but yields

119879119873(0) = 1625 = 064 At the same time there also exists a

significant number of transparent states corresponding to thechain lengths obeying the series 119873lowast = 119865

4+3119896 119896 = 0 1 2

A completely analogous result is obtained for the state 119864 = 0

in the induced transfer model of the binary Fibonacci alloyIn fact making use of the global transfer matrix given by (39)in (69) one gets 119879

119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574

minus1)minus2sin2120581

depending uponwhether119873 is even or odd respectivelyThus


0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)

Figure 22 The quantity 119877(119899) = ln(2120588119899minus 1) is shown as a function of the lattice site 119899 for (a) six-cycle and (b) two-cycle states (from [84]

with permission from Sutherland and Kohmoto copy 1987 by the American Physical Society)

minus4 minus3 minus2 minus1 0 1 2 3 4

E

0

02

04

06

08

1

T

Figure 23 Transmission coefficient as a function of the electronenergy for a 16th generation triadic Cantor lattice All the on-siteenergies are set equal to zero and 119905

119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1

(from [60] with permission from Elsevier)

in this case we obtain a remarkable 119873 parity effect in theresulting transport properties

The properties of electronic states are even more inter-esting in general Fibonacci lattices when one considers thesubset of the energy spectrum formed by those eigenstatessatisfying (40) In this case by plugging the global transfermatrix elements given by (64) into (69) one obtains [49]

119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)

Two important conclusions can be drawn from this expres-sion In the first place the transmission coefficient is alwaysbounded below (ie 119879

119873(119864lowast) = 0) for any lattice length which

proves the truly extended character of the related states Inthe second place since the factor multiplying the sine in thedenominator of expression (73) only vanishes in the trivial(periodic) case 120574 = 1 the critical states we are considering donot verify in general the transparency condition 119879

119873(119864lowast) = 1

in the QP limit

For the sake of illustration consider the extended statesshown in Figure 20 At first sight by comparing both figuresonemay be tempted to think that the transmission coefficientcorresponding to the wave function plotted in Figure 20(a)should be higher than that corresponding to the wave func-tion shown in Figure 20(b) because the charge distributionof the former along the system is more homogeneous thanthat corresponding to the latter Actually however makinguse of expression (73) one finds 119879(119864

lowast

1) = 05909 sdot sdot sdot and

119879(119864lowast

2) = 07425 sdot sdot sdot which is just the opposite caseTherefore

despite the homogeneous distribution of their amplitudesthese estates are not transparent at all

However according to (73) it is also possible to find statessatisfying the transparency condition in general Fibonaccilattices whose lengths satisfy the relationship 119873120601 = 119896120587 119896 =

1 2 which after (65) implies

119864lowast (119896) = plusmnradic1205982 + 4cos2 (119896120587

119873) (74)

In this way these transparent states can be classified accord-ing to a well-defined scheme determined by the integer119896 By combining (40) and (74) we obtain the followingrelationship for the values of the model parameters satisfyingthe transparency condition

120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)

Quite remarkably it was reported that when one choosesthemodel parameters in such away that they satisfy (75) with119896 being a divisor of 119873 (ie 119873119896 = 119901 isin N) one will alwaysget a periodicallymodulated wave functionwhich propagatesballistically through the system [87] The periodicity ofthe long scale modulation is a direct consequence of thedivisibility properties of Fibonacci numbers namely 119865

119896is

a divisor of 119865119901119896

for all 119901 gt 0 [88] For a given systemsize the smaller 119896 is the more periods the envelope ofthe wave functions has and the period of this envelope isexactly119873119896Thus the wave function amplitudes distributionbecomes progressively more homogeneous as the value of


119896 gets smaller as it is illustrated in the different panelsof Figure 24 In that case the wave functions exhibit aremarkable spatial distribution where the QP component(characteristic of short spatial scales) is nicely modulated bya long scale periodic component

On the other hand from (73) we see that among thepossible transparent states there exists a subset satisfyingthe condition 120601 = 0 (120601 = 120587) so that these states aretransparent for any 119873 value As we discussed in Section 33these particular states correspond to the band-edges of theenergy spectrum and they satisfy the condition 120598120574 = plusmn(1 minus

1205742) By comparing with (75) we realize that these band-edge

states correspond to the label 119896 = 0 and 119896 = 119873 (seeFigure 24(a))

Subsequent numerical studies of the energy spectrum ofgeneral Fibonacci lattices have shown that a significant num-ber of electronic states exhibiting transmission coefficientsclose to unity are located around the transparent states givenby (40) [89] An illustrative example is shown in Figure 25 foran energy value close to the resonance energy 119864

lowast= minus1350 =

minus026 By inspecting this plot we notice the existence of twodifferent superimposed structures In fact a periodic-likelong-range oscillation with a typical wavelength of about119873 sites is observed to modulate a QP series of short-rangeminor fluctuations of the wave function amplitude typicallyspreading over a few lattice sites This qualitative descriptionreceives a quantitative support from the study of its Fouriertransform showing two major components in the Fourierspectrum corresponding to the low and high frequencyregions respectively This result suggests that these criticalstates behave in a way quite similar to conventional extendedstates from a physical viewpoint albeit they can not berigorously described in terms of Bloch functions To furtheranalyze this important issue the study of the ac conductivity atzero temperature is very convenient since it is very sensitiveto the distribution nature of eigenvalues and the localizationproperties of the wave function close to the Fermi energy Inthisway by comparing the ac conductivities corresponding toboth periodic and general Fibonacci lattices it was concludedthat the value of the ac conductivity takes on systematicallysmaller values in the Fibonacci case due to the fact that theac conductivity involves the contribution of nontransparentstates within an interval of ℎ120596 around the Fermi level in thiscase [90 91]

In summary although when considering Bloch functionsin periodic systems the notion of extended wave functionscoincides with transparent ones this is no longer true in thecase of fractal and QPS In particular for general Fibonaccisystems in which both diagonal and off-diagonal QP orderare present in theirmodelHamiltonian we have critical stateswhich are not localized (ie 119879

119873(119864lowast) = 0 forall119873 when 119873 rarr

infin) For finite Fibonacci chains one can find transparentstates exhibiting a physical behavior completely analogous tothat corresponding to usual Bloch states in periodic systems(ie 119879

119873(119864lowast) = 1) for a given choice of the model parameters

prescribed by (75) There exists a second class of criticalstates those located close to the transparent ones whichare not strictly transparent (ie 119879

119873(119864) ≲ 1) but exhibit

transmission coefficient values very close to unity Finallythe remaining states in the spectrum show a broad diversityof possible values of the transmission coefficient (ie 0 lt

119879119873(119864) ≪ 1) in agreement with the earlier view of critical

states as intermediate between periodic Bloch wave functions(119879119873(119864) = 1) and localized states (119879

119873(119864) = 0)

43 Electronic States in Koch Fractal Lattices To conclude letus consider the transport properties of electronic states in theKoch lattice This lattice is topologically more complex thanany of the previously considered lattices mainly due to thefact that it has sites with different coordination numbers (seeFigure 15) whereas all sites in Fibonacci triadic Cantor andVicsek fractal lattices are simply twofold coordinated Thishigher topological complexity naturally translates to the PandQ renormalized matrices as well as to their commutatorwhich adopt a more complicated form than those obtainedfor the renormalized matrices in Fibonacci binary alloys thetriadic Cantor lattice and the Vicsek fractal Accordinglythe commutator (51) vanishes in four different cases (i) Thechoice 120582 = 0 reduces the original Koch lattice to a trivialperiodic chain so that all the allowed states minus2 le 119864 le 2are extended (ii) The center of the energy spectrum 119864 = 0corresponds to an extended state [66] (iii) 119864 = plusmnradic2 (iv)The family of states 119864 = minus2120582 For these energies the globaltransfer matrix of the system 119872

119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2

+ 1 can be explicitly evaluated Fromthe knowledge ofM

119896the condition for the considered energy

value to be in the spectrum | tr[M119896]| le 2 can be readily

checked and afterwards its transmission coefficient can beexplicitly determined

Let us consider in the first place the energies 119864 = plusmnradic2In this case we get

tr [M119896] = minus

1

(1 plusmn radic2120582)119899119860minus119899119861

minus (1 plusmn radic2120582)119899119860minus119899119861

(76)

A detailed study of the condition | tr[M119896]| le 2 in (76)

indicates that the only allowed states correspond to 120582 = ∓radic2for which trM

119896= minus2 Consequently these states are just two

particular cases of the more general family (iv) which we willdiscuss next

By taking 119864 = minus2120582 we getQ = minusI where I is the identitymatrix so that M

119896= minusP119899119860 Making use of the Cayley-

Hamilton theorem the global transfer matrix correspondingto the energies 119864 = minus2120582 can be expressed in terms ofChebyshev polynomials of the second kind 119880

119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2

minus 8120582minus4 in the closed form [55]

M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)

where 119902 equiv 2(1205822minus 2)120582

2 From expression (77) we gettr[M119896] = minus2 cos(119899

119860120601) so that we can ensure that these

energies belong to the spectrum in the fractal limit (119896 rarr infin)


0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)

Figure 24 The wave function versus the site number for a Fibonacci lattice with 119873 = 2584 sites 120574 = 2 and 119864lowast(119896) = minus25 cos(120587119896) for

different 119896 values (from [87] with permission from Huang and Gong copy 1998 by the American Physical Society)


0 2584

n

0

1

|120593n|2

Figure 25 Electronic charge distribution corresponding to aperiodic-like state with energy 119864 = minus02601337 for a generalFibonacci lattice with 119873 = 119865

17= 2584 120574 = 15 and 120598 = 01

Its transmission coefficient is 119879119873(119864) ≃ 1ndash10minus15 (from [46] with

permission from CRC Taylor amp Francis group)

The transmission coefficient at a given iteration stage reads[55]

119879119896 (120582) =

1

1 + [(120582 (120582 plusmn 2) 2 (120582 plusmn 1)) sin (119899119860120601)]2 (78)

where the plus (minus) sign in the factor of sin(119899119860120601)

corresponds to the choices 1199051015840 equiv 1 and 1199051015840equiv 119903 = minus1 respec-

tively for the transfer integral of the periodic leads Fromexpression (78) we realize that the transmission coefficientscorresponding to the family (iv) are always bounded belowfor any stage of the fractal growth process which proves theirextended nature in the fractal limit In addition the choices120582 = plusmn2 (119864 = ∓1) correspond to states which are transparentat every stage of the fractal growth process A fact whichensures their transparent nature in the fractal limit as wellFurthermore it is possible to find a number of cross-transferintegral values satisfying the transparency condition 119879

119896= 1

at certain stages of the fractal growth given by the condition119899119860120601 = 119901120587 119901 isin N which allows us to label the different

transparent states at any given stage 119896 in terms of the integer119901

In Figure 26 we plot the transmission coefficient givenby (78) at two successive stages 119896 = 2 and 119896 = 3 as afunction of the cross-transfer value In the first place we notethat the number of 120582 values supporting transparent states ]

120582

progressively increases as the Koch curve evolves toward itsfractal limit according to the power law ]

120582= 2(4

119896minus1+ 1)

It is interesting to compare this figure with the number ofsites 119873 = 4

119896+ 1 present at the stage 119896 in the Koch lattice

Thus we obtain ]120582= (119873 + 3)2 indicating that the number

of Koch lattices able to support transparent states increases

linearly with the system size and consequently that thefractal growth favours the presence of transparent extendedstates in Koch lattices In particular we can state that in thefractal limit there exists an infinitely numerable set of cross-transfer integrals supporting transparent extended states inthe Koch lattice

Another general feature shown in Figure 26 is the pres-ence of a broad plateau around 120582 = 2 where the transmis-sion coefficients take values significantly close to unity Inaddition as 120582 separates from the plateau the local minimaof the transmission coefficient 119879min take on progressivelydecreasing values which tend to zero in the limits 120582 rarr infin

and 120582 rarr 1 This behavior suggests that the best transportproperties in the family (iv) should be expected for thosestates located around the plateau

Up to now we have shown that as the Koch latticeapproaches its fractal limit an increasing number of cross-transfer integrals are able to support transparent states in the119864 = minus2120582 branch of the phase diagram But for a given valueof 120582 how many of the related extended states at an arbitrarystage say 119896 will prevail in the fractal limit 119896 rarr infin Froma detailed analysis of expression (78) it was found that theconsidered states can be classified into two separate classesIn the first class we have those states which are transparentat any stage 119896 In the second class we find states whosetransmission coefficient oscillates periodically between119879 = 1

and a limited range of 119879min = 1 values depending on the valueof the label integer119901 and the fractal growth stage Two generaltrends have been observed in this second class of extendedstates First the values of 119879min are significantly lower forstates corresponding to 119901 lt 0 than for states correspondingto 119901 gt 0 Second at any given fractal stage the values of119879min are substantially higher for states associated with cross-transfer integral values close to the plateau than for statescorresponding to the remaining allowed 120582 values We mustnote however that not all these almost transparent states areexpected to transport inmuch the samemanner as suggestedby the diversity observed in the values of 119879min

In Figure 27 we provide a graphical account of the mostrelevant results obtained in this section In this figurewe showthe phase diagram corresponding to the model Hamiltoniangiven by (48) at the first stage of the fractal process (shadowedlandscape) along with two branches corresponding to thestates belonging to the family 119864 = minus2120582 (thick blacklines) In the way along each symmetrical branch we findthree particular states whose coordinates are respectivelygiven by (plusmn1 ∓2) (plusmnradic2 ∓radic2) and (plusmn2 ∓1) Three of themcorresponding to the choice 120582 gt 0 are indicated by full circleslabelled A B and C in Figure 27 These states correspond totransparent stateswhose transmission coefficients equal unityat every 119896 The remaining states in the branches correspondto almost transparent states exhibiting an oscillating behaviorin their transmission coefficients By comparing Figures26 and 27 we realize that the positions of the transparentstates A-B-C allow us to define three different categories ofalmost transparent states according to their related transportproperties The first class (I) includes those states com-prised between the state A at the border of the spectrum


1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)

Figure 26 Transmission coefficient as a function of the cross-transfer integral at two different stages (a) 119896 = 2 and (b) 119896 = 3 Peaks arelabelled from left to right starting with 119901 = minus4 in (a) Label B corresponds to 119901 = 0 Label C indicates the transparent state at 120582 = 2 (from[55] with permission fromMacia copy 1998 by the American Physical Society)

minus5 5minus5

5

E

120582

A

B

C

Figure 27 Phase diagram showing the Koch lattice spectrum at 119896 =

1 (shadowed areas) and the branches corresponding to the extendedstates family 119864 = minus2120582 (from [55] with permission from Macia copy1998 by the American Physical Society)

and the state B located at a vertex point separating two broadregions of the phase diagram The second class (II) includesthose states comprised between the state B and the state Cclose to the plateau in the transmission coefficient around 120582 =

2 Finally the third class (III) comprises those states beyond

the state C The states exhibiting better transport propertiesbelong to the classes II and III and correspond to those statesgrouping around the plateau near the state C for which thevalues of 119879min are very close to unity As we move apart fromstate C the transport properties of the corresponding almosttransparent states become progressively worse particularlyfor the states belonging to the class III for which values of119879min as low as 10minus3 can be found

5 Outlook and Perspectives

In the preceding sections we have presented results obtainedin the study of wave functions in fractal and QPS by mainlyconsidering two complementary criteria namely their spa-tial structure and the value of their related transmissioncoefficients In the case of periodic crystals or amorphousmaterials both criteria are simply related to each otherThus in periodic crystals we have Bloch states exhibiting aperiodically extended spatial structure and 119879

119873(119864) = 1 for

any 119873 value Therefore the allowed states in the spectrumare both extended and transparent wave functions On theother hand uncorrelated random systems have exponentiallylocalized wave functions with 119879

119873(119864) = 0 Thus all states

in these systems are both localized and have vanishingtransmission coefficients These simple scenarios no longerapply to either fractal or QPS having a richer class of wavefunctions exhibiting a plethora of amplitude distributionpatternsThese patterns must be generally described in termsof multifractal analysis and exhibit a broad diversity oftransmission coefficient values spanning the full range 0 lt

119879119873(119864) le 1


Two points should be highlighted regarding the transmis-sion coefficient limiting values (i) since 119879

119873(119864) = 0 in general

strictly localized states are not present in QPS (ii) the verypossibility of having 119879

119873(119864) = 1 for some particular energy

values indicates that the notion of transparent state must bewidened to include eigenstates which are not Bloch functionsat all

Additionally fractal systems generally possess an evenricher variety of wave functions stemming from their inher-ent combination of self-similarity (a property shared withQPS) and fractal dimensionality (manifested by the pres-ence of sites with different coordination numbers along thechain)

The algebraic conditions R119899119894= plusmnI along with [R

119894R119895] = 0

where R119894denotes a suitable renormalized transfer matrix

allow one to properly pick up certain special energy valuesrelated to wave functions which exhibit quite homogeneousamplitude distributions generally lacking distinctive self-similar features The absence of self-similar arrangementsof peaks can be traced back to the fact that for theseenergies the global transfer matrix for the original fractalor QP lattice can be expressed in terms of an effectiveperiodic chain (eg triadic Cantor lattice) or two effectiveperiodic chains (eg general Fibonacci lattices) upon appro-priate decimationrenormalization of the original chainsIn both cases such a possibility naturally leads to thepresence of periodic-like distributions of the wave functionsamplitudes

By keeping in mind the main results reviewed in thiswork concerning the spatial structure of the wave functionsbelonging to fractal and QPS it could be concluded that therecent trend of abandoning the former use of ldquocritical wavefunctionrdquo to generically refer to all eigenstates belonging tosingular continuous spectra should be encouraged [92] Infact whereas the electronic states found in the Aubry-Andreself-dual model can be properly regarded as being strictlycritical ones (in the sense that they undergo ametal-insulatorcritical transition) this criterion no longer holds for the statesfound in both fractal and QPS albeit all of them displaymultifractal properties In our opinion the fundamentalproperty in order to properly classify the nature of statesin aperiodic systems must focus on their related transportproperties rather than on the wave function amplitudesdistribution patterns The very existence of transparent (bal-listic propagation) states in these systems provides conclusiveevidence on the unsuitability of the dichotomist view in termsof Bloch and exponentially localized states in order to accountfor the electronic transport properties in aperiodic systemsAccordingly we would like to tentatively propose the use ofthe term singular wave functions to generically refer to thebroad family of eigenstates belonging to aperiodic systemsexhibiting singular continuous energy spectra

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 The theory of these functions was developed by HaraldBohr (1887ndash1951) brother of the well-known physicistNiels Bohr [93]

2 As an illustrative counter example we can consider thefunction

119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)

which is AP but not QP since its Fourier spectrum doesnot have a finite basis

3 Though exponentially localized decaying states are con-sidered as representative of most localization processesin solid state physics one must keep in mind that otherpossible mathematical functions are also possible to thisend

4 The tight-binding approximation considers that both thepotential and the electronic wave functions are sharplypeaked on the atomic sites

5 We note that (9) is invariant under the simultaneoustransformation

120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)

This property shows that the spectra corresponding toopposite values of the potential aremirror images of eachother with respect to the center of the spectrum

6 For the sake of simplicity one can adopt units such thatℎ = 2119898 without loss of generality

7 Note that all of the transfer integrals take on the samevalue

8 The existence of exponentially localized wave functionshas been recently reported from numerical studies of(10) with the bichromatic potential 119881(119909) = 2120582

1cos119909 +

21205822cos120572119909 with rational 120572 = 119901119902 corresponding to

certain specific energy values (119864 gt 2(1205821+ 1205822)) This

system then suggests the possible presence of a pure-point spectral component not restricted to low energyvalues in certain periodic systems [44]

9 This equation is also referred to as Harperrsquos equation(see Section 16) or the almost Mathieu operator whichresults from the discretization of the classical Mathieuequation

1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)

where 1198870and 1198871are real constants

10 The details of the crossover between localized andextended regimes have been studied in detail for somespecial values of irrational 120572 values only so that onemayask whether these particular 120572 values are representativeenough

11 That is 120572 = lim119898rarrinfin

(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers


12 By comparing (14) and (20) we realize that in Harperrsquosequation the potential strength maximum value is con-strained to the value120582 = 2 from the onset whereas in theAubry-Andre equation this potential strengthmaximumis one of the free parameters of the model

13 Note that this discontinuous character excludes thepossibility of an absolutely continuous spectral measure

14 Quite interestingly this description written in the pre-fractal era uses the term ldquoclusteringrdquo instead of ldquosplittingrdquoin order to describe the energy spectrum fragmentedpattern This semantics seems to be more appropriatewhen considering the physical origin of the hierarchicalpatterns in terms of resonance effects involving a seriesof neighboring building blocks interspersed through thelattice

15 The choice of binary lattices composed of just twodifferent types of atoms is just a matter of mathematicalsimplicity though one should keep inmind that all of thethermodynamically stable QCs synthesized during theperiod 1987ndash2000 were ternary compounds In 2000 thediscovery of the first binary QCwas reported (belongingto the (Cd Ca)Yb system) and its atomic structure wassolved in 2007 [7]

16 To the best of our knowledge this state was first studiedby Kumar [50] who apparently confused it with the 119864 =

0 eigenstate of the Fibonacci standard transfer modelshown in Figure 13(d)

17 According to the Cayley-Hamilton theorem any 119899 times 119899

square matrix over the real or complex field is a root ofits own characteristic polynomial [94] Making use ofthis property ifM belongs to the 119878119897(2R) group one canreadily obtain by induction the expression

M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)

with 120593 equiv cosminus1119911 and 119911 equiv (12) trM being Chebyshevpolynomials of the second kind satisfying the recursionrelation

119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911

18 In an analogous way the condition B3 = plusmnB reducesthe global transfer matrix of the original triadic Cantorlattice to that corresponding to a periodic structure witha unit cell119860119861 (plus an isolated119860 at the right end) On thecontrary the condition B3 = plusmnA transforms the originalglobal transfer matrix into that corresponding to thenonhomogeneous fractal lattice 119860119861119860311986111986051198611198603119861119860 sdot sdot sdot

19 A more distinctive lattice-like structure has beenreported for the wave function amplitudes distributionof certain states in Thue-Morse lattices [74]

20 Note that in the critical wave functions considered in theAubry-Andre model the self-similar property extendsover the entire support of the function rather than ona limited interval of it

21 Note that for the global transfer matrices of the form

(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)

one obtains the eigenvalues 120582plusmn= plusmn1

22 Note that (72) is invariant under the inversion operation119905119861rarr 119905minus1

119861

Acknowledgments

It is with great pleasure that the author thanks ProfessorArunava Chakrabarti for many insightful conversations anduseful comments on the nature of states in aperiodic systemsduring the last two decades Heshe also thank ProfessorLuca Dal Negro Professor UichiroMizutani Professor DavidDamanik Professor Roland Ketzmerick and Dr StefanieThiem for sharing with himher useful information Theauthor thanks Victoria Hernandez for a careful criticalreading of the paper

References

[1] D Bohm ldquoQuantum theory as an indication of a new orderin physicsmdashpart A The development of new orders as shownthrough the history of physicsrdquo Foundations of Physics vol 1no 4 pp 359ndash381 1971

[2] D Shechtman I Blech D Gratias and J W Cahn ldquoMetallicphase with long-range orientational order and no translationalsymmetryrdquoPhysical Review Letters vol 53 no 20 pp 1951ndash19531984

[3] D Levine and P J Steinhardt ldquoQuasicrystals a new class ofordered structuresrdquo Physical Review Letters vol 53 no 26 pp2477ndash2480 1984

[4] ldquoICrU Report of the Executive Committee for 1991rdquo ActaCrystallographica A vol 48 p 922 1992

[5] T Janssen G Chapuis and M de Boissieu Aperiodic CrystalsFrom Modulated Phases to Quasicrystals Oxford UniversityPress Oxford UK 2007

[6] M Baake Quasicrystals An Introduction to Structure PhysicalProperties and Applications edited by J B Suck M Schreiberand P Haussler Springer Berlin Germany 2002

[7] W Steurer and S Deloudi Crystallography of QuasicrystalsmdashConcepts Methods and Structures vol 126 of Springer Series inMaterials Science Springer Berlin Germany 2009

[8] S Ostlund and R Pandit ldquoRenormalization-group analysisof the discrete quasiperiodic Schrodinger equationrdquo PhysicalReview B vol 29 no 3 pp 1394ndash1414 1984

[9] U Mizutani Introduction to the Electron Theory of MetalsCambridge University Press Cambridge UK 2001

[10] R del Rio S Jitomirskaya Y Last and B Simon ldquoWhat islocalizationrdquo Physical Review Letters vol 75 no 1 pp 117ndash1191995

[11] D H Dunlap H L Wu and P W Phillips ldquoAbsence oflocalization in a random-dimermodelrdquo Physical Review Lettersvol 65 no 1 pp 88ndash91 1990


[12] A Sanchez E Macia and F Domınguez-Adame ldquoSuppressionof localization in Kronig-Penney models with correlated disor-derrdquo Physical Review B vol 49 p 147 1994

[13] S Ostlund R Pandit D Rand H J Schellnhuber and E DSiggia ldquoOne-dimensional Schrodinger equation with an almostperiodic potentialrdquo Physical Review Letters vol 50 no 23 pp1873ndash1876 1983

[14] N Mott andW D Twose ldquoThe theory of impurity conductionrdquoAdvances in Physics vol 10 p 107 1961

[15] E Macia F Domınguez-Adame and A Sanchez ldquoEnergyspectra of quasiperiodic systems via information entropyrdquoPhysical Review E vol 50 p 679 1994

[16] J Bellissard A Bovier and J M Ghez Differential Equationswith Application to Mathematical Physics edited byW F AmesEMHarrell II and J V Herod Academic Press BostonMassUSA 1993

[17] D Damanik M Embree and A Gorodetski ldquoSpectral proper-ties of Schrodinger operators arising in the study of quasicrys-talsrdquo Tech Rep TR12-21 Rice University CAAM Department2012

[18] L Dal Negro N Lawrence J Trevino and GWalsh ldquoAperiodicorder for nanophotonicsrdquo in Optics of Aperiodic StructuresFundamentals and Device Applications Pan Stanford BocaRaton Fla USA 2014

[19] J Bellissard B Iochum and D Testard ldquoContinuity propertiesof the electronic spectrum of 1D quasicrystalsrdquo Communica-tions in Mathematical Physics vol 141 no 2 pp 353ndash380 1991

[20] J Bellissard A Bovier and J M Ghez ldquoGap labelling theoremsfor one dimensional discrete schrodinger operatorsrdquo Reviews inMathematical Physics vol 4 p 1 1992

[21] J M Luck ldquoCantor spectra and scaling of gap widths indeterministic aperiodic systemsrdquo Physical Review B vol 39 no9 pp 5834ndash5849 1989

[22] E Macia ldquoThe role of aperiodic order in science and technol-ogyrdquo Reports on Progress in Physics vol 69 p 397 2006

[23] M Dulea M Johansson and R Riklund ldquoLocalization of elec-trons and electromagnetic waves in a deterministic aperiodicsystemrdquo Physical Review B vol 45 no 1 pp 105ndash114 1992

[24] M Dulea M Johansson and R Riklund ldquoTrace-map invariantand zero-energy states of the tight-binding Rudin-Shapiromodelrdquo Physical Review B vol 46 no 6 pp 3296ndash3304 1992

[25] M Dulea M Johansson and R Riklund ldquoUnusual scaling ofthe spectrum in a deterministic aperiodic tight-bindingmodelrdquoPhysical Review B vol 47 no 14 pp 8547ndash8551 1993

[26] A Suto ldquoSingular continuous spectrum on a cantor set of zeroLebesgue measure for the Fibonacci Hamiltonianrdquo Journal ofStatistical Physics vol 56 no 3-4 pp 525ndash531 1989

[27] J Bellissard B Iochum E Scoppola and D Testard ldquoSpectralproperties of one dimensional quasi-crystalsrdquo Communicationsin Mathematical Physics vol 125 no 3 pp 527ndash543 1989

[28] B Iochum and D Testard ldquoPower law growth for the resistancein the Fibonaccimodelrdquo Journal of Statistical Physics vol 65 no3-4 pp 715ndash723 1991

[29] J Bellissard A Bovier and J-M Ghez ldquoSpectral properties ofa tight binding Hamiltonian with period doubling potentialrdquoCommunications in Mathematical Physics vol 135 no 2 pp379ndash399 1991

[30] A Bovier and J M Ghez ldquoSpectral properties of one-dimensional Schrodinger operators with potentials generatedby substitutionsrdquoCommunications inMathematical Physics vol158 p 45 1993 Erratum in Communications in MathematicalPhysics vol 166 pp 431 1994

[31] A Bovier and J-M Ghez ldquoRemarks on the spectral propertiesof tight-binding and Kronig-Penney models with substitutionsequencesrdquo Journal of Physics A vol 28 no 8 pp 2313ndash23241995

[32] S Aubry and S Andre ldquoAnalyticity breaking and Andersonlocalization in in commensurate latticesrdquo Annals of the IsraelPhysical Society vol 3 p 133 1980

[33] H Aoki ldquoDecimationmethod of real-space renormalization forelectron systems with application to random systemsrdquo PhysicaA vol 114 no 1ndash3 pp 538ndash542 1982

[34] H Aoki ldquoCritical behaviour of extended states in disorderedsystemsrdquo Journal of Physics C vol 16 no 6 pp L205ndashL208 1983

[35] C M Soukoulis and E N Economou ldquoFractal character ofeigenstates in disordered systemsrdquo Physical Review Letters vol52 no 7 pp 565ndash568 1984

[36] C Castellani and L Peliti ldquoMultifractal wavefunction at thelocalisation thresholdrdquo Journal of Physics A vol 19 no 8 articleno 004 pp L429ndashL432 1986

[37] M Schreiber ldquoFractal character of eigenstates in weakly disor-dered three-dimensional systemsrdquoPhysical ReviewB vol 31 no9 pp 6146ndash6149 1985

[38] H Aoki ldquoFractal dimensionality of wave functions at themobility edge quantum fractal in the Landau levelsrdquo PhysicalReview B vol 33 no 10 pp 7310ndash7313 1986

[39] Y Oono T Ohtsuki and B Kramer ldquoInverse participationnumber and fractal dimensionality of electronic states in a twodimensional system in strong perpendicular magnetic fieldrdquoJournal of the Physical Society of Japan vol 58 pp 1705ndash17161989

[40] H Grussbach and M Schreiber ldquoMultifractal electronic wavefunctions in the Anderson model of localizationrdquo ModernPhysics Letters vol 6 p 851 1992

[41] D R Hofstadter ldquoEnergy levels and wave functions of Blochelectrons in rational and irrational magnetic fieldsrdquo PhysicalReview B vol 14 no 6 pp 2239ndash2249 1976

[42] M Kohmoto ldquoLocalization problem and mapping of one-dimensional wave equations in random and quasiperiodicmediardquo Physical Review B vol 34 no 8 pp 5043ndash5047 1986

[43] S N Evangelou ldquoMulti-fractal spectra and wavefunctions ofone-dimensional quasi-crystalsrdquo Journal of Physics C vol 20p L295 1987

[44] S Mugabassi and A Vourdas ldquoLocalized wavefunctions inquantum systems with multiwell potentialsrdquo Journal of PhysicsA vol 43 Article ID 325304 2010

[45] EMacia and FDomınguez-Adame ldquoThree-dimensional effectson the electronic structure of quasiperiodic systemsrdquo Physica Bvol 216 p 53 1995

[46] EMacia BarberAperiodic Structures inCondensedMatter CRCTaylor amp Francis Boca Raton Fla USA 2009

[47] F Domınguez-Adame E Macia B Mendez C L Roy and AKhan ldquoFibonacci superlattices of narrow-gap IIIndashV semicon-ductorsrdquo Semiconductor Science and Technology vol 10 p 7971995

[48] M Kohmoto B Sutherland and C Tang ldquoCritical wavefunctions and a Cantor-set spectrum of a one-dimensionalquasicrystal modelrdquo Physical Review B vol 35 no 3 pp 1020ndash1033 1987

[49] E Macia and F Domınguez-Adame ldquoPhysical nature of criticalwave functions in Fibonacci systemsrdquo Physical Review Lettersvol 76 p 2957 1996


[50] V Kumar ldquoExtended electronic states in a Fibonacci chainrdquoJournal of Physics vol 2 no 5 article no 026 pp 1349ndash13531990

[51] R Rammal ldquoNature of eigenstates on fractal structuresrdquo Physi-cal Review B vol 28 no 8 pp 4871ndash4874 1983

[52] W A Schwalm and M K Schwalm ldquoElectronic propertiesof fractal-glass modelsrdquo Physical Review B vol 39 pp 12872ndash12882 1989

[53] W A Schwalm and M K Schwalm ldquoExplicit orbits forrenormalization maps for Green functions on fractal latticesrdquoPhysical Review B vol 47 no 13 pp 7847ndash7858 1993

[54] R F S Andrade and H J Schellnhuber ldquoElectronic stateson a fractal exact Greenrsquos-function renormalization approachrdquoPhysical Review B vol 44 no 24 pp 13213ndash13227 1991

[55] E Macia ldquoElectronic transport in the Koch fractal latticerdquoPhysical Review B vol 57 no 13 pp 7661ndash7665 1998

[56] P Kappertz R F S Andrade andH J Schellnhuber ldquoElectronicstates on a fractal inverse-iteration methodrdquo Physical Review Bvol 49 no 20 pp 14711ndash14714 1994

[57] Z Lin and M Goda ldquoPower-law eigenstates of a regular Vicsekfractal with hierarchical interactionsrdquoPhysical ReviewB vol 50no 14 pp 10315ndash10318 1994

[58] A Chakrabarti ldquoField induced delocalization in a Koch fractalrdquoPhysical Review B vol 60 no 15 pp 10576ndash10579 1999

[59] A Chakrabarti ldquoExact results for infinite and finite Sierpinskigasket fractals extended electron states and transmission prop-ertiesrdquo Journal of Physics vol 8 no 50 pp 10951ndash10957 1996

[60] S Sengupta A Chakrabarti and S Chattopadhyay ldquoElectronicproperties of a Cantor latticerdquo Physica B vol 344 no 1ndash4 pp307ndash318 2004

[61] A Chakrabarti and B Bhattacharyya ldquoAtypical extended elec-tronic states in an infinite Vicsek fractal an exact resultrdquoPhysical Review B vol 54 no 18 pp R12625ndashR12628 1996

[62] A Chakrabarti and B Bhattacharyya ldquoSierpinski gasket in amagnetic field electron states and transmission characteristicsrdquoPhysical Review B vol 56 no 21 pp 13768ndash13773 1997

[63] C S Jayanthi and S Y Wu ldquoFrequency spectrum of a generalVicsek fractalrdquo Physical Review B vol 48 no 14 pp 10188ndash10198 1993

[64] Y E Levy and B Souillard ldquoSuperlocalization of electrons andwaves in fractal mediardquo Europhysics Letters vol 4 p 233 1987

[65] S Sengupta A Chakrabarti and S Chattopadhyay ldquoElectronictransport in a Cantor stub waveguide networkrdquo Physical ReviewB vol 72 no 13 Article ID 134204 2005

[66] R F S Andrade and H J Schellnhuber ldquoExact treatment ofquantum states on a fractalrdquo Europhysics Letters vol 10 p 731989

[67] Y Gefen B B Mandelbrot and A Aharony ldquoCritical phenom-ena on fractal latticesrdquo Physical Review Letters vol 45 no 11 pp855ndash858 1980

[68] E Macia ldquoExploiting aperiodic designs in nanophotonicdevicesrdquo Reports on Progress in Physics vol 75 Article ID036502 2012

[69] L Dal Negro and S V Boriskina ldquoDeterministic aperiodicnanostructures for photonics and plasmonics applicationsrdquoLaser and Photonics Reviews vol 6 no 2 pp 178ndash218 2012

[70] E Macia ldquoThe importance of being aperiodic optical devicesrdquoin Optics of Aperiodic Structures Fundamentals and DeviceApplications Pan Stanford Boca Raton Fla USA 2014

[71] ZValyVardeny ANahata andAAgrawal ldquoOptics of photonicquasicrystalsrdquo Nature Photonics vol 7 p 177 2013

[72] S V Gaponenko Introduction to Nanophotonics CambridgeUniversity Press Cambridge UK 2010

[73] M Severin M Dulea and R Riklund ldquoPeriodic and quasiperi-odic wavefunctions in a class of one-dimensional quasicrystalsan analytical treatmentrdquo Journal of Physics vol 1 no 45 pp8851ndash8858 1989

[74] C S Ryu G Y Oh and M H Lee ldquoExtended and critical wavefunctions in aThue-Morse chainrdquo Physical Review B vol 46 no9 pp 5162ndash5168 1992

[75] W Gellermann M Kohmoto B Sutherland and P C TaylorldquoLocalization of light waves in Fibonacci dielectric multilayersrdquoPhysical Review Letters vol 72 no 5 pp 633ndash636 1994

[76] A Chakrabarti S N Karmakar and R K MoitraldquoRenormalization-group analysis of extended electronicstates in one-dimensional quasiperiodic latticesrdquo PhysicalReview B vol 50 no 18 pp 13276ndash13285 1994

[77] V Kumar and G Ananthakrishna ldquoElectronic structure of aquasiperiodic superlatticerdquo Physical Review Letters vol 59 no13 pp 1476ndash1479 1987

[78] X C Xie and S Das Sarma ldquoExtended electronic states in aFibonacci superlatticerdquo Physical Review Letters vol 60 no 15p 1585 1988

[79] S Sil S N Karmakar R K Moitra and A ChakrabartildquoExtended states in one-dimensional lattices application to thequasiperiodic copper-mean chainrdquo Physical Review B vol 48no 6 pp 4192ndash4195 1993

[80] M Kohmoto and J R Banavar ldquoQuasiperiodic lattice elec-tronic properties phonon properties and diffusionrdquo PhysicalReview B vol 34 no 2 pp 563ndash566 1986

[81] Q Niu and F Nori ldquoRenormalization-group study of one-dimensional quasiperiodic systemsrdquo Physical Review Lettersvol 57 no 16 pp 2057ndash2060 1986

[82] QNiu andFNori ldquoSpectral splitting andwave-function scalingin quasicrystalline and hierarchical structuresrdquo Physical ReviewB vol 42 no 16 pp 10329ndash10341 1990

[83] G J Jin and Z D Wang ldquoAre self-similar states in Fibonaccisystems transparentrdquo Physical Review Letters vol 79 p 52981997

[84] B Sutherland and M Kohmoto ldquoResistance of a one-dimensional quasicrystal power-law growthrdquo Physical ReviewB vol 36 no 11 pp 5877ndash5886 1987

[85] S Das Sarma and X C Xie ldquoConductance fluctuations in one-dimensional quasicrystalsrdquo Physical Review B vol 37 no 3 pp1097ndash1102 1988

[86] J M Dubois Useful Quasicrystals World Scientific Singapore2005

[87] XHuang andCGong ldquoProperty of Fibonacci numbers and theperiodiclike perfectly transparent electronic states in Fibonaccichainsrdquo Physical Review B vol 58 no 2 pp 739ndash744 1998

[88] D Kalman and R Mena ldquoThe Fibonacci numbers exposedrdquoMathematics Magazine vol 76 pp 167ndash181 2003

[89] R Oviedo-Roa L A Perez and C Wang ldquoAc conductivity ofthe transparent states in Fibonacci chainsrdquo Physical Review Bvol 62 no 21 pp 13805ndash13808 2000

[90] V Sanchez L A Perez R Oviedo-Roa and C Wang ldquoRenor-malization approach to theKubo formula in Fibonacci systemsrdquoPhysical Review B vol 64 Article ID 174205 2001

[91] V Sanchez and C Wang ldquoApplication of renormalizationand convolution methods to the Kubo-Greewood formula inmultidimensional Fibonacci systemsrdquo Physical Review B vol70 Article ID 144207 2004


[92] SThiemandM Schreiber ldquoWavefunctions quantumdiffusionand scaling exponents in golden-mean quasiperiodic tilingsrdquoJournal of Physics vol 25 Article ID 075503 2013

[93] H Bohr Collected Mathematical WorksmdashII Almost PeriodicFunctions Dansk Matematisk Forening Copenhagen Den-mark 1952

[94] F R GantmacherTheTheory of Matrices 2 Chelsea New YorkNY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


Disorder Order

Random Periodic

(a) Before 1992

Order

Periodic Aperiodic

(b) After 1992

Amorphous Crystals Classicalcrystals

Aperiodiccrystals

Incommensuratecomposites

QuasicrystalsIncommensuratemodulated

phases

3

44 5 6

gt3

Figure 1 In 1992 the notion of crystal was widened beyondmere periodicity This conceptual diagram presents the positionof aperiodic crystals no longer based on the notion of periodictranslation symmetry among the different orderings of matter Thediverse aperiodic crystal families are arranged according to thedimension 119863 (see (2)) of their embedding hyperspaces (numericallabels)

P

AP

QP

Figure 2 Graphical representation of the hierarchical nestedrelationship among almost periodic (AP) quasiperiodic (QP) andperiodic (P) functions

the Fourier transform in order to properly analyze atomicdensity distributions

Despite the fact that more than two decades have elapsedsince the crystal notion has been properly revisited one canstill find in the literature a lot of works plainly stating thatquasiperiodic systems (QPS) provide an example of inter-mediate structures between ordered and disordered systemsSentences like this certainly rely on a too vague notion of theterm ldquointermediaterdquo which apparently ignores the fact thatevery QP function can be expressed in terms of a numerableset of periodic functions in an appropriate high-dimensionalspace Accordingly periodic functions are but the simplerparticular instances of the more general QP ones From thisperspective QPS are not only perfectly ordered structuresbut theymay even be regarded as having a higher order degreethan periodic ones This viewpoint is nicely illustrated bythe hierarchical relationship between almost periodic (AP)QP and periodic functions shown in Figure 2 Indeed froma mathematical viewpoint periodic functions are a specialcase of QP functions which are in turn a special case of APfunctions1

Therefore rather than adopting the old dichotomist way(which only allows one to get fuzzy qualitative comparisonsas to whether a particular system is less random or moreperiodic than any other one) it may be more fruitful to thinkin terms of the different hierarchies of order to which thesesystems belong (see Figure 7)

Almost periodic functions can be uniformly approxi-mated by Fourier series containing a countable infinity ofpairwise incommensurate reciprocal periods (frequencies)[5 6] When the set of reciprocal periods (frequencies)required can be generated fromafinite-dimensional basis theresulting function is referred to as a QP one2 For the sake ofillustration let us consider an aperiodic crystal whose atomicdistribution is given by a QP function expressed in terms ofits discrete Fourier decomposition

119891 (x) = sum

k119886k119890119894ksdotx

(1)

where the reciprocal vectors are defined by

k =

119873

sum

119895=1

119899119895b119895 (2)

where b119895are reciprocal lattice basis vectors If the minimal

number of these basis vectors is larger than three that is119873 gt 3 in (2) then a higher dimensional description is neededto describe the reciprocal lattice and the related structure isan aperiodic crystal Otherwise we obtain a periodic crystal(Figure 1(b))

The simplest one-dimensional example of a QP functioncan be written as

119891 (119909) = 1198601cos (119909) + 119860

2cos (120572119909) (3)

where 120572 is an irrational number and 1198601and 119860

2are real

numbers It is interesting to note that this QP function can beobtained as a one-dimensional projection of a related periodicfunction in two dimensions

119891 (119909 119910) = 1198601cos119909 + 119860

2cos119910 (4)

through the restriction 119910 = 120572119909 This property is at the basisof the so-called cut and project method which is widely usedin the study of QCs In fact since any QP function can bethought of as deriving from a periodic function in a spaceof higher dimension most of the basic notions of classicalcrystallography can be properly extended to the study of QCsin appropriate hyperspaces [5 7]

12 Extended Localized and Critical Wave Functions Oncewe have clarified that aperiodic crystals do not occupy avague intermediate position between periodic crystal andamorphous matter representatives it is pertinent to indicatethat there exists a physical context in which one can properlytalk about the existence of an intermediate state betweenorder and disorder This scenario is that occurring whena system undergoes a phase transition from solid to liquidstates experiencing critical fluctuations at all scales This


0 2583

Sites n

Wav

e fun

ctio

n120595n

(a)

minus233 0 233

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

(b)

minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

(c)






0

x

uk(x)

minusa2 a2

(a)

x

minus5a 0 5a

eikxuk(x)

(b)







ℎ2

2119898

1198892120595

1198891199092+ [119864 minus 119881 (119909)] 120595 = 0 (5)


plusmn(119909) = 119860




120595 (119909) = 119906119896 (119909) 119890

119894119896119909 (6)



119896(119909) for






120595 (119909 + 119899119886) = 119906119896 (119909 + 119899119886) 119890

119894119896(119909+119899119886)

= 119906119896 (119909) 119890

119894119896119909119890119894119896119899119886

= 119906119896 (119909) 119890

119894119896119909= 120595 (119909)

(7)




120595119899 (119909) = 119860

119899119890minusℓ119899|119909| (8)














119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= (119864 minus 119881119899) 120595119899 (9)




1cos(119899 + ]) + 120582

2cos(119899120572 + ])







119899andor 119905

119899119899plusmn1


119899and

119905119899119899plusmn1




119899119899+1 sequences









minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0 987

Sites n

Sites n

minus987

minus233 0 233

(a)

(b)

(c)








119860= 120592119861





119899= 120582 cos(119899120572 + ])) can take on




Fibonacci 119860 119861119892 (119860) = 119860119861

119892(119861) = 119860

Silver mean 119860 119861119892(119860) = 119860119860119861

119892(119861) = 119860

Bronze mean 119860 119861119892(119860) = 119860119860119860119861

119892(119861) = 119860

Precious means 119860 119861119892(119860) = 119860

119899119861

119892(119861) = 119860

Copper mean 119860 119861119892(119860) = 119860119861119861

119892(119861) = 119860

Nickel mean 119860 119861119892(119860) = 119860119861119861119861

119892(119861) = 119860

Metallic means 119860 119861119892(119860) = 119860119861

119899

119892(119861) = 119860

Mixed means 119860 119861119892(119860) = 119860

119899119861119898

119892(119861) = 119860

Thue-Morse 119860 119861119892(119860) = 119860119861

119892(119861) = 119861119860

Period-doubling 119860 119861119892(119860) = 119860119861

119892(119861) = 119860119860


119892(119860) = 119860119862

119892(119861) = 119860

119892(119862) = 119861

Rudin-Shapiro 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119860119862

119892(119862) = 119863119861

119892(119863) = 119863119862

Paper folding 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119862119861

119892(119862) = 119860119863

119892(119863) = 119862119863



[minus1198892

1198891199092+ 119881 (119909)]120595 = 119864120595 (10)



(119867120595)119899= 119905120595119899+1

+ 119905120595119899minus1

+ 120582120599 (119899) 120595119899 119899 isin Z (11)




119875) absolutely


120583 = 120583119875cup 120583AC cup 120583SC (12)





0 20 40 60 80 100


0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()


0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50





Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling


Ideal quasicrystal

Amorphous matter


120583P 120583AC120583SC

120583P

120583AC

120583SC















[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 2583

Sites n

Wav

e fun

ctio

n120595n

(a)

minus233 0 233

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

(b)

minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

(c)






0

x

uk(x)

minusa2 a2

(a)

x

minus5a 0 5a

eikxuk(x)

(b)







ℎ2

2119898

1198892120595

1198891199092+ [119864 minus 119881 (119909)] 120595 = 0 (5)


plusmn(119909) = 119860




120595 (119909) = 119906119896 (119909) 119890

119894119896119909 (6)



119896(119909) for






120595 (119909 + 119899119886) = 119906119896 (119909 + 119899119886) 119890

119894119896(119909+119899119886)

= 119906119896 (119909) 119890

119894119896119909119890119894119896119899119886

= 119906119896 (119909) 119890

119894119896119909= 120595 (119909)

(7)




120595119899 (119909) = 119860

119899119890minusℓ119899|119909| (8)














119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= (119864 minus 119881119899) 120595119899 (9)




1cos(119899 + ]) + 120582

2cos(119899120572 + ])







119899andor 119905

119899119899plusmn1


119899and

119905119899119899plusmn1




119899119899+1 sequences









minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0 987

Sites n

Sites n

minus987

minus233 0 233

(a)

(b)

(c)








119860= 120592119861





119899= 120582 cos(119899120572 + ])) can take on




Fibonacci 119860 119861119892 (119860) = 119860119861

119892(119861) = 119860

Silver mean 119860 119861119892(119860) = 119860119860119861

119892(119861) = 119860

Bronze mean 119860 119861119892(119860) = 119860119860119860119861

119892(119861) = 119860

Precious means 119860 119861119892(119860) = 119860

119899119861

119892(119861) = 119860

Copper mean 119860 119861119892(119860) = 119860119861119861

119892(119861) = 119860

Nickel mean 119860 119861119892(119860) = 119860119861119861119861

119892(119861) = 119860

Metallic means 119860 119861119892(119860) = 119860119861

119899

119892(119861) = 119860

Mixed means 119860 119861119892(119860) = 119860

119899119861119898

119892(119861) = 119860

Thue-Morse 119860 119861119892(119860) = 119860119861

119892(119861) = 119861119860

Period-doubling 119860 119861119892(119860) = 119860119861

119892(119861) = 119860119860


119892(119860) = 119860119862

119892(119861) = 119860

119892(119862) = 119861

Rudin-Shapiro 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119860119862

119892(119862) = 119863119861

119892(119863) = 119863119862

Paper folding 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119862119861

119892(119862) = 119860119863

119892(119863) = 119862119863



[minus1198892

1198891199092+ 119881 (119909)]120595 = 119864120595 (10)



(119867120595)119899= 119905120595119899+1

+ 119905120595119899minus1

+ 120582120599 (119899) 120595119899 119899 isin Z (11)




119875) absolutely


120583 = 120583119875cup 120583AC cup 120583SC (12)





0 20 40 60 80 100


0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()


0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50





Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling


Ideal quasicrystal

Amorphous matter


120583P 120583AC120583SC

120583P

120583AC

120583SC















[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

x

uk(x)

minusa2 a2

(a)

x

minus5a 0 5a

eikxuk(x)

(b)







ℎ2

2119898

1198892120595

1198891199092+ [119864 minus 119881 (119909)] 120595 = 0 (5)


plusmn(119909) = 119860




120595 (119909) = 119906119896 (119909) 119890

119894119896119909 (6)



119896(119909) for






120595 (119909 + 119899119886) = 119906119896 (119909 + 119899119886) 119890

119894119896(119909+119899119886)

= 119906119896 (119909) 119890

119894119896119909119890119894119896119899119886

= 119906119896 (119909) 119890

119894119896119909= 120595 (119909)

(7)




120595119899 (119909) = 119860

119899119890minusℓ119899|119909| (8)














119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= (119864 minus 119881119899) 120595119899 (9)




1cos(119899 + ]) + 120582

2cos(119899120572 + ])







119899andor 119905

119899119899plusmn1


119899and

119905119899119899plusmn1




119899119899+1 sequences









minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0 987

Sites n

Sites n

minus987

minus233 0 233

(a)

(b)

(c)








119860= 120592119861





119899= 120582 cos(119899120572 + ])) can take on




Fibonacci 119860 119861119892 (119860) = 119860119861

119892(119861) = 119860

Silver mean 119860 119861119892(119860) = 119860119860119861

119892(119861) = 119860

Bronze mean 119860 119861119892(119860) = 119860119860119860119861

119892(119861) = 119860

Precious means 119860 119861119892(119860) = 119860

119899119861

119892(119861) = 119860

Copper mean 119860 119861119892(119860) = 119860119861119861

119892(119861) = 119860

Nickel mean 119860 119861119892(119860) = 119860119861119861119861

119892(119861) = 119860

Metallic means 119860 119861119892(119860) = 119860119861

119899

119892(119861) = 119860

Mixed means 119860 119861119892(119860) = 119860

119899119861119898

119892(119861) = 119860

Thue-Morse 119860 119861119892(119860) = 119860119861

119892(119861) = 119861119860

Period-doubling 119860 119861119892(119860) = 119860119861

119892(119861) = 119860119860


119892(119860) = 119860119862

119892(119861) = 119860

119892(119862) = 119861

Rudin-Shapiro 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119860119862

119892(119862) = 119863119861

119892(119863) = 119863119862

Paper folding 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119862119861

119892(119862) = 119860119863

119892(119863) = 119862119863



[minus1198892

1198891199092+ 119881 (119909)]120595 = 119864120595 (10)



(119867120595)119899= 119905120595119899+1

+ 119905120595119899minus1

+ 120582120599 (119899) 120595119899 119899 isin Z (11)




119875) absolutely


120583 = 120583119875cup 120583AC cup 120583SC (12)





0 20 40 60 80 100


0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()


0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50





Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling


Ideal quasicrystal

Amorphous matter


120583P 120583AC120583SC

120583P

120583AC

120583SC















[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= (119864 minus 119881119899) 120595119899 (9)




1cos(119899 + ]) + 120582

2cos(119899120572 + ])







119899andor 119905

119899119899plusmn1


119899and

119905119899119899plusmn1




119899119899+1 sequences









minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0 987

Sites n

Sites n

minus987

minus233 0 233

(a)

(b)

(c)








119860= 120592119861





119899= 120582 cos(119899120572 + ])) can take on




Fibonacci 119860 119861119892 (119860) = 119860119861

119892(119861) = 119860

Silver mean 119860 119861119892(119860) = 119860119860119861

119892(119861) = 119860

Bronze mean 119860 119861119892(119860) = 119860119860119860119861

119892(119861) = 119860

Precious means 119860 119861119892(119860) = 119860

119899119861

119892(119861) = 119860

Copper mean 119860 119861119892(119860) = 119860119861119861

119892(119861) = 119860

Nickel mean 119860 119861119892(119860) = 119860119861119861119861

119892(119861) = 119860

Metallic means 119860 119861119892(119860) = 119860119861

119899

119892(119861) = 119860

Mixed means 119860 119861119892(119860) = 119860

119899119861119898

119892(119861) = 119860

Thue-Morse 119860 119861119892(119860) = 119860119861

119892(119861) = 119861119860

Period-doubling 119860 119861119892(119860) = 119860119861

119892(119861) = 119860119860


119892(119860) = 119860119862

119892(119861) = 119860

119892(119862) = 119861

Rudin-Shapiro 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119860119862

119892(119862) = 119863119861

119892(119863) = 119863119862

Paper folding 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119862119861

119892(119862) = 119860119863

119892(119863) = 119862119863



[minus1198892

1198891199092+ 119881 (119909)]120595 = 119864120595 (10)



(119867120595)119899= 119905120595119899+1

+ 119905120595119899minus1

+ 120582120599 (119899) 120595119899 119899 isin Z (11)




119875) absolutely


120583 = 120583119875cup 120583AC cup 120583SC (12)





0 20 40 60 80 100


0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()


0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50





Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling


Ideal quasicrystal

Amorphous matter


120583P 120583AC120583SC

120583P

120583AC

120583SC















[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus4181 0 4181

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0 987

Sites n

Sites n

minus987

minus233 0 233

(a)

(b)

(c)








119860= 120592119861





119899= 120582 cos(119899120572 + ])) can take on




Fibonacci 119860 119861119892 (119860) = 119860119861

119892(119861) = 119860

Silver mean 119860 119861119892(119860) = 119860119860119861

119892(119861) = 119860

Bronze mean 119860 119861119892(119860) = 119860119860119860119861

119892(119861) = 119860

Precious means 119860 119861119892(119860) = 119860

119899119861

119892(119861) = 119860

Copper mean 119860 119861119892(119860) = 119860119861119861

119892(119861) = 119860

Nickel mean 119860 119861119892(119860) = 119860119861119861119861

119892(119861) = 119860

Metallic means 119860 119861119892(119860) = 119860119861

119899

119892(119861) = 119860

Mixed means 119860 119861119892(119860) = 119860

119899119861119898

119892(119861) = 119860

Thue-Morse 119860 119861119892(119860) = 119860119861

119892(119861) = 119861119860

Period-doubling 119860 119861119892(119860) = 119860119861

119892(119861) = 119860119860


119892(119860) = 119860119862

119892(119861) = 119860

119892(119862) = 119861

Rudin-Shapiro 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119860119862

119892(119862) = 119863119861

119892(119863) = 119863119862

Paper folding 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119862119861

119892(119862) = 119860119863

119892(119863) = 119862119863



[minus1198892

1198891199092+ 119881 (119909)]120595 = 119864120595 (10)



(119867120595)119899= 119905120595119899+1

+ 119905120595119899minus1

+ 120582120599 (119899) 120595119899 119899 isin Z (11)




119875) absolutely


120583 = 120583119875cup 120583AC cup 120583SC (12)





0 20 40 60 80 100


0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()


0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50





Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling


Ideal quasicrystal

Amorphous matter


120583P 120583AC120583SC

120583P

120583AC

120583SC















[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Fibonacci 119860 119861119892 (119860) = 119860119861

119892(119861) = 119860

Silver mean 119860 119861119892(119860) = 119860119860119861

119892(119861) = 119860

Bronze mean 119860 119861119892(119860) = 119860119860119860119861

119892(119861) = 119860

Precious means 119860 119861119892(119860) = 119860

119899119861

119892(119861) = 119860

Copper mean 119860 119861119892(119860) = 119860119861119861

119892(119861) = 119860

Nickel mean 119860 119861119892(119860) = 119860119861119861119861

119892(119861) = 119860

Metallic means 119860 119861119892(119860) = 119860119861

119899

119892(119861) = 119860

Mixed means 119860 119861119892(119860) = 119860

119899119861119898

119892(119861) = 119860

Thue-Morse 119860 119861119892(119860) = 119860119861

119892(119861) = 119861119860

Period-doubling 119860 119861119892(119860) = 119860119861

119892(119861) = 119860119860


119892(119860) = 119860119862

119892(119861) = 119860

119892(119862) = 119861

Rudin-Shapiro 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119860119862

119892(119862) = 119863119861

119892(119863) = 119863119862

Paper folding 119860 119861 119862119863

119892(119860) = 119860119861

119892(119861) = 119862119861

119892(119862) = 119860119863

119892(119863) = 119862119863



[minus1198892

1198891199092+ 119881 (119909)]120595 = 119864120595 (10)



(119867120595)119899= 119905120595119899+1

+ 119905120595119899minus1

+ 120582120599 (119899) 120595119899 119899 isin Z (11)




119875) absolutely


120583 = 120583119875cup 120583AC cup 120583SC (12)





0 20 40 60 80 100


0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()


0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50





Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling


Ideal quasicrystal

Amorphous matter


120583P 120583AC120583SC

120583P

120583AC

120583SC















[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 20 40 60 80 100


0

20

40

60

80

Four

ier c

oeffi

cien

ts|F()|

Four

ier c

oeffi

cien

ts|F()|

s ()

0 20 40 60 80 100

s ()

0 20 40 60 80 100

s ()


0

10

20

30

40

50

Four

ier c

oeffi

cien

ts|F()|

0

10

20

30

40

50





Ener

gy sp

ectr

um

Usual crystalline

matter

Spiral lattice

Fibonacci

Period-doubling


Ideal quasicrystal

Amorphous matter


120583P 120583AC120583SC

120583P

120583AC

120583SC















[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics













[minus1198892

1198891199092+ 120582 cos (2120587120572 + ])]120595 = 119864120595 (13)





1 determines





120595119899minus1

+ 120595119899+1





120595119899= 1198901198942120587119896119899

infin

sum

119898=minusinfin

119886119898119890119894119898(2120587119899120572+]) (15)


119886119898+1

+ 119886119898minus1

+ [119864 minus cos (2120587119899120572 + ])] 119886119898 = 0 (16)


=4

120582 119864 =

2119864

120582 ] = 119896 (17)





119899




[1

119901 ln120572ln

119878119898

119878119898+119901

] 119878119898equiv

1

119902119898

119902119898

sum

119899=minus119902119898

10038161003816100381610038161205951198991003816100381610038161003816

2 (18)


0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 2583

Sites n

Wav

e fun

ctio

n120595n

120582 = 15 120581 = 14 1206010 = 01234 (1205950 1205951) = (1 0)

(a)

0 1

x

Hul

l fun

ctio

n 120594

(x)

0 1

x

Hul

l fun

ctio

n 120594

(x)

(b)


119899= 1198901198941198991205872







120573 equiv

0 120582 lt 2

minus (0639 plusmn 0005) 120582 = 2

minus1 120582 gt 2

(19)











minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus4181 0 4181

Sites n

Sites n

Sites n

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

0

1

Mag

nitu

de o

f wav

e fun

ctio

n|120595

n|

minus987 0 987

minus233 0 233

(a)

(b)

(c)







120595119899minus1

+ 120595119899+1

+ [119864 minus 2 cos (2120587119899120572 + ])] 120595119899 = 0 (20)


















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



















= 119901119899119902119899






1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1 2 3 4 5 6 7 8 9

n

minus8

minus7

minus6

minus5

minus4

minus3

E(e

V)



Elsevier)





119899119899+1= 119905 forall119899)


120595119899+1

+ 120595119899minus1

= (119864 minus 119881119899) 120595119899 (21)





119905119899119899+1

120595119899+1

+ 119905119899119899minus1

120595119899minus1

= 119864120595119899 (22)

06

065

07

075

E(e

V)

E(e

V)

E(e

V)

066

067

068

069

0672

0674

0676

0678

n = 3 n = 6 n = 9




119860119861= 119905119861119860

and 119905119860119860



119899119899+1


119899 sequence





(

120595119899+1

120595119899

) = (

119864 minus 119881119899

119905119899119899+1

minus119905119899119899minus1

119905119899119899+1

1 0

)(

120595119899

120595119899minus1

) equiv M119899(

120595119899

120595119899minus1

)

(23)






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






(a)


t

(b)

V



(c)


V = 0

(d)


119899119899plusmn1




119860equiv 120598 = minus119881

119861119905119860119860

= 119905119860119861

equiv 1


119861119905119860119860

= 119905119860119861

equiv 1


119860= 1 119905

119861= 1


119860119860= 119905119860119861

equiv 1


1and 120595


(120595119873+1

120595119873

) =

1

prod

119899=119873

M119899(1205951

1205950

) equiv M119873 (119864) (

1205951

1205950

) (24)






1 0) B equiv (

119864 + 120598 minus1

1 0) (25)


119860= 120598 = minus119881

119861(Table 3) The




1 0) D equiv (

119864 minus119905119861

1 0)



119861

1 0

)

(26)


C K equiv (

119864 minus120574

1 0) L equiv (

119864120574minus1

minus120574minus1

1 0

) (27)

where 120574 equiv 119905119860119860


119860119861equiv 1


A B Z equiv (

119864 minus 120598 minus120574

1 0) Y equiv (

120574minus1


1 0

)

(28)


119899119899+1







1198651= 1 119865

7= 21 119865

13= 377 119865

19= 6765

1198652= 2 119865

8= 34 119865

14= 610 119865

20= 10946

1198653= 3 119865

9= 55 119865

15= 987 119865

21= 17711

1198654= 5 119865

10= 89 119865

16= 1597 119865

22= 28657

1198655= 8 119865

11= 144 119865

17= 2584 119865

23= 46368

1198656= 13 119865

12= 233 119865

18= 4181 119865

24= 75025


M119873 (119864) =

1

prod

119899=119873



119899+1= M119899minus1


0= B

and M1

= A so that if 119873 = 119865119895 where 119865



+119865119895minus2

with1198651= 1 and 119865


119860= 119865119895minus1




M119873 (119864) =

1

prod

119899=119873


(30)



119861(1198642minus 1199052

119861minus119864

119864 minus1

) (31)


119860= 119865119895minus2


119861= 119865119895minus3





M119873 (119864) =

1

prod

119899=119873



(32)


S119861equiv C2 = (

1198642minus 1 minus119864

119864 minus1)

S119860equiv KLC = 120574

minus1(

119864(1198642minus 1205742minus 1) 120574

2minus 1198642

1198642minus 1 minus119864

)

(33)



119860=

119865119895minus3


matrices S119861







M119873 (119864) =

1

prod

119899=119873


119861R119860R119860R119861R119860 (34)





119860= 119865119895minus3

matrices R119860

and119899119861= 119865119895minus4


matrices

R119861= (


120598 + 119864 minus1

)

R119860= 120574minus1

(

11987711 (119864) 120574

2minus (119864 minus 120598)

2


)

(35)

with 11987711(119864) = (119864 + 120598)[(119864 minus 120598)







[R119860R119861] =

120598 (1 + 1205742) minus 119864 (1 minus 120574

2)

120574(

1 0

119864 + 120598 minus1)

[S119860 S119861] =

119864 (1205742minus 1)

120574(1 0

119864 minus1)

[A B] =1199052

119861minus 1

119905119861

(0 1

1 0) [AB] = 2120598 (

0 1

1 0)

(36)


119860= 119881119861equiv 0 or 119905

119860119860= 119905119860119861

equiv 1


S119861 (0) = (

minus1 0


119860 (0) = (

0 120574

minus120574minus1

0

) (37)


M119873 (0) = S119899119861

119861(0) S119899119860119860

(0)

= (minus1)119899119861 (

minus119880119899119860minus2 (0) 120574119880

119899119860minus1 (0)


119899119860minus2 (0)

)

(38)


119873(0)| = |2119880

119899119860minus2(0)| =




119899 and 1198802119899+1


119860= 119865119895minus3

integer namely

M119873 (0) =

(minus1)119899119861I 119899

119860even

(minus1)119899119861S119860 (0) 119899

119860odd

(39)


0= 1205951= 1) The 119860 sites have


119899|2= 1 or 1205742



119864lowast= 120598

1 + 1205742

1 minus 1205742 (40)











119860119861119860 997888rarr 119860119861119860119861119861119861119860119861119860

997888rarr 119860119861119860119861119861119861119860119861119860119861119861119861119861119861119861119861119861119861119860119861119860119861119861119861119860119861119860


(41)




M3 (119864) = ABA

M9 (119864) = ABAB3ABA


(42)



12

43

5 1 3 5t

t t

(I) (II)

(III)

k = 1

k = 2

1205980

1205980

1205980 12059801205981

1205981

1205981

1205981

1205981

120598112059821205980

tN = 3

N = 9


0



B3 = (

1198803 (119909) minus119880

2 (119909)

1198802 (119909) minus119880

1 (119909)) equiv (

plusmn1 0

0 plusmn1) (43)

where 1198803(119909) = 8119909

3minus 4119909 119880

2(119909) = 4119909

2minus 1 and 119909 =



119896









1205981= 1205980+

21199052

119864 minus 1205980

1205982= 1205981+

21199052


1minus 1199052 (119864 minus 120598

0))

(44)




2= 1205981




V119896= (

119864 minus 120598119896

119905minus1

1 0

) (45)


[V119896+1

V119896] =

120598119896+1

minus 120598119896

119905(0 1

1 0) (46)




M3 (119864) = V

0V1V0

M9 (119864) = V

0V1V0V0V2V0V0V1V0


M27 (119864) = V



M3119896+1 (119864) = M

3119896minus1 (119864)M

3119896+1 (119864)M

3119896minus1 (119864)

(47)



119867 = sum

119899

|119899⟩ ⟨119899 + 1| + |119899⟩ ⟨119899 minus 1| + 120582119891 (119899)


(48)


119891 (119899) = 120575 (0 119899) +

119896minus1

sum

119904

120575(4119904

2 119899 (mod 4

119904)) (49)




119896 can



Pminus1M119896= M2

119896minus1QM2

119896minus1 (50)



[PQ] =

120582119864 (1198642minus 2) (2 + 120582119864)

1199033


2) 119903 119903

2

(1 minus 1198642) (1198642minus 3) (119864

2minus 2) 119903

)

(51)


119860R119861]






1198892119864

1198891199092+ [

1205962

11988821198992(119909) minus 119896

2

]E = 0 (52)







minus2minus1

0

12

4

6

8

h

(a)


G H G H G H G HF F

minus7 minus1 1 7

(b)



Z

(a)

Z

n(Z)

nB

nA

(b)

Z

V(z)

(c)







5= 243



0 50 100 150 200

Site

0

05

1

15

2

25

3|a

mpl

itude

|

(a)

0 50 100 150 200

Site

0

05

1

15

2

25

3

|am

plitu

de|

(b)


119860= 0 119881

119861= minus1 and









M119865119895+1

(119864) = M119865119895minus1

(119864)M119865119895

(119864) 119895 ge 2 (53)


1198971

+ 1198651198972

+ sdot sdot sdot + 119865119897119894


119894isin N


M119899 (119864) = M

119865119897119894

(119864) sdot sdot sdotM1198651198972

(119864)M1198651198971

(119864) (54)


M1 (0) = B (0) = (

0 minus1

1 0)

M2 (0) = A (0) = minus(

119905119861

0

0 119905minus1

119861

)

(55)


M3 (0) = M

1 (0)M2 (0) = (0 119905minus1

119861

minus119905119861

0)

M5 (0) = M

2 (0)M3 (0) = (0 minus1

1 0) = M

1 (0)

M8 (0) = M

3 (0)M5 (0) = (119905minus1

1198610

0 119905119861

)

M13 (0) = M

5 (0)M8 (0) = (0 minus119905

119861

119905minus1

1198610

)

M21 (0) = M

8 (0)M13 (0) = (0 minus1

1 0) = M

1 (0)

M34 (0) = M

13 (0)M21 (0) = minus(119905119861

0

0 119905minus1

119861

) = M2 (0)

(56)


21(0)M

34(0) = M

1(0)M

2(0) =



(0) = M119865119895



1 17711

n

n

0

32

64

|120595n|

|120595n|

minus987 987

|120595n|

minus233 233

n

n

minus55 55

|120595n|

(a)

(b)

(c)

(d)







119861times 120595







M16 (0) = M

3 (0)M13 (0) = (

119905minus2

1198610

0 1199052

119861

)

M71 (0) = M

3 (0)M13 (0)M55 (0) = M16 (0)M3 (0)

= (

0 119905minus3

119861

minus1199053

1198610

)


M304 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)

= M71 (0)M13 (0) = (

119905minus4

1198610

0 1199054

119861

)

M1291 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

= M304 (0)M3 (0) = (

0 119905minus5

119861

minus1199055

1198610

)

M5472 (0) = M

3 (0)M13 (0)M55 (0)M233 (0)M987 (0)

timesM4181 (0) = M

1291 (0)M13 (0) = (

119905minus6

1198610

0 1199056

119861

)

(57)


(0) = M119865119895


119899lowast

=

1199052119901

1198611205950and 120595

119899lowast

= minus1199052119901+1






119899 997888rarr 119899120591minus3

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905minus1

119861

10038161003816100381610038161205951003816100381610038161003816 gt 1

10038161003816100381610038161205951003816100381610038161003816 997888rarr

10038161003816100381610038161205951003816100381610038161003816 119905119861

10038161003816100381610038161205951003816100381610038161003816 lt 1

(58)


119899≃ 119899120585 with 120585 = | ln(119905

119861119905119860)| ln 120591

3 (for 119905119861

= 2

and 119905119860


(119905minus2119898

1198610

0 1199052119898

119861

) (59)





1 17711

n

|120595n|












20= 10946 and 119865







R119861= (

1199022minus 1 minus119902120574

119902120574minus1

minus1

)

R119860= (

119902 (1199022minus 2) 120574 (1 minus 119902

2)

120574minus1

(1199022minus 1) minus119902

)

(60)

where

119902 equiv2120598120574

1 minus 1205742 (61)





Mplusmn

119873(119864lowast) = (plusmn1)

119899119860R119899119861119861

= (plusmn1)119899119860 (

119880119899119861minus2

(minus1

2) ∓120574119880

119899119861minus1

(minus1

2)

∓120574minus1119880119899119861minus1

(minus1

2) 119880

119899119861

(minus1

2)

)

(62)



119860= R3 where

R = (

119902 minus120574

120574minus1

0

) (63)


M119873(119864lowast) = R119873 = (

119880119873

minus120574119880119873minus1

120574minus1119880119873minus1


) (64)


and

119909 =119902

2=

1



= 2119879119899


119873(119864lowast)] = 2 cos(119873120601)




chain with119873 = 11986516



lowast









119899(1) = 119899+1 and119880


119899(119899+1)


M119873(119864+

lowast) = (

119873 + 1 minus120574119873

120574minus1119873 1 minus 119873

)

M119873(119864minus

lowast) = (minus1)

119873(

119873 + 1 120574119873


)

(66)

where 119864+




0 1597

n

0

02

04

06

08

1

12

|120593n|2

(a)

0 1597

n

0

02

04

06

08

1

12

|120593n|2

(b)


lowast

1= minus54 and (b) 120574 = 2

120598 = 05 and 119864lowast




minus2 minus1 1 2

minus3

minus2

minus1

0

0

1

2

3






120588119873 (119864) =

1

4(1198722

11+1198722

12+1198722

21+1198722

22minus 2) (67)



119873(119864) le 120588

0119873120585 with






⟨120588119873⟩ equiv

1

]

]

sum

119894=1

120588119873(119864119894) (68)





119879119873 (119864)

=4(detM

119873)2sin2120581

[11987212

minus11987221

+ (11987211

minus11987222) cos 120581]2 + (119872

11+11987222)2sin2120581

(69)







120588119873 (119864) =

1 minus 119879119873 (119864)

119879119873 (119864)

(70)





119873(119864) = 1 so



119860= 0 119881

119861= minus1 119905 = 1 1205981015840 = 0 and 119905

1015840= 1 reads

119879119873 (0) = [1 +

1

3sin2 (4 (119896 minus 1)

120587

3)]

minus1

(71)












119879119873 (0) =

41199052

119861

(1 + 1199052

119861)2 (72)








4+3119896 119896 = 0 1 2



119873(0) = 1 or 119879

119873(0) = 4(120574 + 120574




0 400

n

0

R(n)

36ln(120601)

(a)

0 400

n

0

R(n)

36ln(120601)

(b)




E

0

02

04

06

08

1

T


119871= 1 119905

119878= 2 1205981015840 = 0 and 119905

1015840= 1




119879119873(119864lowast) = [

[

1 +

(1 minus 1205742)2

(4 minus 1198642lowast) 1205742

sin2 (119873120601)]

]

minus1

(73)




119873(119864lowast) = 1

in the QP limit


lowast


119879(119864lowast






119873) (74)


120598 = plusmn1 minus 1205742

120574cos(119896120587

119873) (75)


119896is

a divisor of 119865119901119896








lowast= minus1350 =







119873(119864) ≲ 1) but exhibit




119873(119864) = 0)


119896= P119899119860Q119899119861 with 119899

119860=

4119896minus1

+ 1 and 119899119861= 4119896minus2







1



(76)








119899(119909) with 119909 equiv

cos120601 = minus1 + 8120582minus2


M119896= (

119880119899119860minus1

minus 119880119899119860

minus119902119880119899119860minus1

119902119880119899119860minus1

119880119899119860minus2

minus 119880119899119860minus1

) (77)






0 800 1600 2400

N = 2584 k = 2584

Number of sites

|120593n(k)|2

(au

)

(a)

N = 2584 k = 1292

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(b)

N = 2584 k = 646

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(c)

N = 2584 k = 323

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(d)

N = 2584 k = 152

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(e)

N = 2584 k = 136

0 800 1600 2400

Number of sites

|120593n(k)|2

(au

)

(f)




0 2584

n

0

1

|120593n|2


17= 2584 120574 = 15 and 120598 = 01




119879119896 (120582) =

1





119896= 1




120582


120582= 2(4

119896minus1+ 1)












1 10

120582

0

04

08

12

t

B

C

(a)

1 10

120582

0

04

08

12

t

(b)


minus5 5minus5

5

E

120582

A

B

C








119873(119864) = 1 for


119873(119864) = 0 Thus all states


119879119873(119864) le 1



119873(119864) = 0 in general






119894R119895] = 0






Endnotes



119891 (119909) = lim119873rarrinfin

119873

sum

119899=1

1

119899sin(

119909

2119899) (79)





120595119899997888rarr (minus1)

119899120595119899 119881

119899997888rarr minus119881

119899 119864 997888rarr minus119864 (80)





1cos119909 +





1198892119891

1198891199092+ (1198870+ 1198871cos119909)119891 (81)




(119901119898119902119898) where 119901

119898and 119902

119898are

coprime integers










M119873 = 119880119873minus1 (119911)M minus 119880

119873minus2 (119911) I (82)

where

119880119873equivsin (119873 + 1) 120593

sin120593 (83)


119880119899+1

minus 2119911119880119899+ 119880119899minus1

= 0 119899 ge 1 (84)

with 1198800(119911) = 1 and 119880

1(119911) = 2119911





(0 119905minus(2119898+1)

119861

1199052119898+1

1198610

) (85)



119861

Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Review Article On the Nature of Electronic Wave Functions