Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry

Mikhail S. Plyushchay1,2,* and Luis-Miguel Nieto2,†

1Departamento de Fısica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile2Departamento de Fısica Teorica, Atomica y Optica, Universidad de Valladolid, 47071, Valladolid, Spain

(Received 15 July 2010; published 24 September 2010)

We study supersymmetry of a self-isospectral one-gap Poschl-Teller system in the light of a mirror

symmetry that is based on spatial and shift reflections. The revealed exotic, partially broken, nonlinear

supersymmetry admits seven alternatives for a grading operator. One of its local, first order supercharges

may be identified as a Hamiltonian of an associated one-gap, nonperiodic Bogoliubov-de Gennes system.

The latter possesses a nonlinear supersymmetric structure, in which any of the three nonlocal generators of

a Clifford algebra may be chosen as the grading operator. We find that the supersymmetry generators for

both systems are the Darboux-dressed integrals of a free spin-1=2 particle in the Schrodinger picture, or of

a free massive Dirac particle. Nonlocal Foldy-Wouthuysen transformations are shown to be involved in

the supersymmetric structure.

DOI: 10.1103/PhysRevD.82.065022 PACS numbers: 11.30.Pb, 03.65.�w, 11.10.Kk, 11.10.Lm

I. INTRODUCTION

A Z2 grading structure lies in the basis of super-symmetry. In the early years of supersymmetric quantummechanics [1,2], Gendenshtein and Krive observed [3] thatin some systems the Z2 grading may be provided by areflection operator. The origin of such a hidden supersym-metric structure [4–6] was explained recently in [7] bymeans of a Foldy-Wouthuysen transformation for thecase of a linear supersymmetry that is based on the firstorder Darboux transformations [8] and is described by theLie superalgebraic relations.

Braden and Macfarlane [9], and in a more broad contextDunne and Feinberg [10] revealed that a linear N ¼ 2supersymmetric extension of the periodic finite-gapquantum systems may produce completely isospectralsystems characterized by the same, but a shifted potential.The name self-isospectrality was coined by the latterauthors for such a phenomenon, which was studied laterby Fernandez et al. [11] as Darboux displacements, seealso [12].

The both periodic and nonperiodic finite-gap quantumsystems, being related to nonlinear integrable systems [13],find many important applications in diverse areas of phys-ics, ranging from condensed matter physics, QCD andcosmology, to the string theory [14–25].

A higher order generalization of the Darboux trans-formations, known as the Darboux-Crum transformations[8], gives rise to a higher derivative generalization ofsupersymmetric quantum mechanics [26], characterizedby nonlinear superalgebraic relations [5,27–29].

Soon after the discovery of the self-isospectrality, it wasfound that in some periodic finite-gap systems this phe-nomenon may be associated with not a linear, but nonlinear

supersymmetry [30]. Later on, hidden nonlinear supersym-metry [5] was revealed in unextended finite-gap periodicfinite-gap systems [31]. It was also established that self-isospectral n-gap periodic systems with a half-period shiftare described by a special nonlinear supersymmetricstructure that includes a hidden supersymmetry of theorder 2nþ 1, whose local generator, being a Lax operator,factorizes into the Darboux intertwining operators of theexplicit nonlinear, of order 2k, k � 1, and linear or non-linear, of order 2ðn� kÞ þ 1, supersymmetries [32].There is an essential difference between supersym-

metries of the periodic and nonperiodic self-isospectralfinite-gap systems. In the former case, linear N ¼ 2 super-symmetry generators, as a part of a broader structure, mayannihilate two states of zero energy, while they cannot havezero modes in the nonperiodic case. A little attention wasgiven, however, to the study of the self-isospectralityphenomenon in the nonperiodic finite-gap systems.In the present paper, we investigate the interplay of the

self-isospectrality, reflections, Darboux transformations,and nonlinear and hidden supersymmetries for nonperiodicfinite-gap quantum systems. This is done here for thesimplest case of a one-gap, self-isospectral reflectionlessPoschl-Teller (PT) system, and an associated one-gapBogoliubov-de Gennes (BdG) system that is described bya first order Hamiltonian.1 We reveal a rich supersymmet-ric structure, related to several admissible choices of thegrading operator (seven for PT and three for BdG) in theserelated systems. Our analysis is based on a mirror symme-try that includes a free particle as an essential element. Wefind that all the nontrivial integrals are a Darboux-dressed

*mikhail.plyushchay@usach.cl†luismi@metodos.fam.cie.uva.es

1The BdG system [33] appears in many physical problems,including, particularly, superconductivity theory, fractional fer-mion number, the Peierls effect, and the crystalline condensatesin the chiral Gross-Neveu and Nambu-Jona Lasinio models, see[14–19,34–36].

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form of the corresponding integrals of a free spin-1=2particle system, and show that nonlocal Foldy-Fouthuysen transformations are involved in the exoticsupersymmetric structure.

The paper is organized as follows. In the next section, amirror symmetry of the self-isospectral, one-gap reflec-tionless PT system is discussed, and its local and nonlocalintegrals of motion are identified via a Darboux dressing ofa free particle. In Sec. III we analyze the eigenstates of thethree basic local integrals. Nonlinear superalgebraic struc-ture and its peculiarities are described in Sec. IV. In Sec. Vwe show that the unextended, single one-gap PT systemmay be characterized by an exotic hidden nonlinear super-symmetry, which is related to supersymmetry of theextended, self-isospectral system by a nonlocal Foldy-Wouthuysen transformation. In Sec. VI, identifying oneof the local supercharges of the self-isospectral PT systemas a ð1þ 1ÞD Dirac Hamiltonian, we describe the non-linear supersymmetry of the associated one-gap BdG sys-tem. Section VII is devoted to the discussion of the results.

II. MIRROR SYMMETRYAND INTEGRALS OFMOTION OF SELF-ISOSPECTRAL ONE-GAP PT

SYSTEM

Consider a one-gap, nonperiodic reflectionless PT sys-tem [8,37],2

H1 ¼ � d2

dx2� 2cosh�2xþ 1; (2.1)

and factorize the Hamiltonian,

H1 ¼ AAy; A ¼ d

dx� tanhx: (2.2)

It is connected with a (shifted for a constant) free particleHamiltonian,

H0 ¼ AyA ¼ � d2

dx2þ 1; (2.3)

by the intertwining relations,

AH0 ¼ H1A; H0Ay ¼ AyH1: (2.4)

The PT system (2.1) is almost isospectral to the system(2.3). The eigenstates of the same energy, H1c

E1 ¼ Ec E

1 ,H0c

E0 ¼ Ec E

0 , are related3 by a Darboux transformation

c E1 ðxÞ ¼ Ac E

0 ðxÞ; c E0 ðxÞ ¼ Ayc E

1 ðxÞ; (2.5)

and the spectra are in one-to-one correspondence exceptone bound, square integrable state of zero energy, which is

missing in the free particle spectrum. The explicit form ofthe PT eingestates is

E ¼ 0: �0ðxÞ ¼ 1

coshx; E ¼ 1: �1ðxÞ ¼ � tanhx;

(2.6)

E ¼ 1þ k2 > 1: c�kðxÞ ¼ ð�ik� tanhxÞe�ikx;

k > 0:(2.7)

The doublet states of the continuous part of the spectrum(E> 1) are obtained from the plane wave states e�ikx,the singlet state �1 corresponds to a singlet state c 1

0 ¼ 1(k ¼ 0) of the free particle. A nonphysical state c 0

0 ¼sinhx, which is a formal eigenstate of H0, is mapped tothe unique bound singlet state �0 in the PT system, �0 ¼Ac 0

0. The latter is a zero mode of the first order operator

Ay, Ay�0 ¼ 0. There is one energy gap in the spectrum ofthe reflectionless PT system that separates a zero energyeigenvalue of the bound state from the continuous part ofthe spectrum (k � 0).Let us shift the coordinate x for þ� and for �� (� > 0),

and denote

A� ¼ d

dx� tanhðxþ �Þ; A�� ¼ d

dx� tanhðx� �Þ;

H� ¼ A�Ay� ; H�� ¼ A��A

y��:

As the PT systemH� is just theH�� translated for 2�, thesetwo Hamiltonians are completely isospectral.The systems H� and H�� are related by a mirror (with

respect to x ¼ 0) symmetry,

RH� ¼ H��R; (2.8)

where R is a spatial reflection operator, Rx ¼ �xR,R2 ¼ 1. The reflection R intertwines, therefore, the twoisospectral PT systems, cf. (2.4). It also intertwines thefactorizing operators,

RA� ¼ �A��R; RAy� ¼ �Ay��R: (2.9)

In addition, we introduce a reflection operator4 for theshift parameter �, T � ¼ ��T , T 2 ¼ 1, which also inter-twines the Hamiltonians and the factorizing operators,

T H� ¼ H��T ; (2.10)

T A� ¼ A��T ; T Ay� ¼ Ay��T : (2.11)

Each of the shifted Hamiltonians,H� andH��, may alsobe treated as a mirror image of another, with a free particlesystem playing the role of the mirror. Indeed, a shiftof x does not change the free particle Hamiltonian (2.3),

2The Hamiltonian of the reflectionless one-gap system of themost general form is H1 ¼ �d2=dx2 � 2�2cosh�2�ðx� x0Þ þconst; we put here � ¼ 1, const ¼ 1, and fixed, for the moment,x0 ¼ 0.

3Up to constant, energy-dependent factors which are of noimportance for us here.

4From a viewpoint of an associated free Dirac particle system,see below, T may be treated as a kind of a charge conjugationoperator.

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H0 ¼ Ay�A� ¼ Ay��A��, and we get the two different sets

of intertwining relations,

A�H0 ¼ H�A�; H0Ay� ¼ Ay

�H�; (2.12)

A��H0 ¼ H��A��; H0Ay�� ¼ Ay��H��: (2.13)

Combining them, we find the second order operators thatgenerate a Darboux-Crum transform between the twomutually shifted PT systems,

Y�H� ¼ H��Y�; Y��H�� ¼ H�Y��; (2.14)

where

Y� ¼ A��Ay� ; Yy

� ¼ Y��: (2.15)

The mirror H0 is present virtually here by means of rela-tions (2.12) and (2.13),

Y�H� ¼ A��ðAy�H�Þ ¼ A��ðH0A

y� Þ ¼ ðA��H0ÞAy

�

¼ ðH��A��ÞAy� ¼ H��Y�: (2.16)

The Darboux-Crum intertwining relations (2.14) aretranslated into the language of supersymmetric quantummechanics. Consider the composed system described bythe diagonal two-by-two Hamiltonian

H ¼ H� 00 H��

� �; (2.17)

and define the matrix operators

Q1 ¼ 0 Yy�

Y� 0

!; Q2 ¼ i�3Q1: (2.18)

Because of (2.14), the Q1 and Q2 are the integrals ofmotion of the extended system (2.17), ½H ; Qa� ¼ 0, a ¼1, 2. The diagonal Pauli matrix �3 can be taken as agrading operator, � ¼ �3, �

2 ¼ 1. Then H and Qa areidentified, respectively, as bosonic and fermionic opera-tors, ½�;H � ¼ 0, f�; Qag ¼ 0. With taking into accountEqs. (2.12) and (2.13), one finds that the supercharges Qa

generate a nonlinear, second order superalgebra

fQa;Qbg ¼ 2�abH 2: (2.19)

The system (2.17), being a one-gap (superextended)self-isospectral reflectionless system, possesses other non-trivial integrals.5 To find them, we use the following ob-servation [39]. Suppose that some Hamiltonians,6 H and~H, are related by the intertwining identities DH ¼ ~HD,HDy ¼ Dy ~H, where D is a differential operator of anyorder. If J is an integral of the system H, then DJDy is theintegral of the system ~H,

½J;H� ¼ 0 ) ½~J; ~H� ¼ 0; ~J ¼ DJDy: (2.20)

Associate with the system (2.17) an extended system

H 0 ¼ H0 00 H0

� �; (2.21)

composed from the two copies of the free particle.The systems (2.17) and (2.21) are related, in correspon-dence with (2.12) and (2.13), by the identities DH 0 ¼HD, H 0Dy ¼ DyH , where the matrix intertwiningoperator is

D ¼ A� 00 A��

� �: (2.22)

According to (2.20), the supercharges (2.18) of the super-extended PT system (2.17) correspond to the trivial, spinintegrals �1 and �2 of the free particle system (2.21).Other, ‘‘dressed’’ integrals can be found in a similar way.They are displayed in Table I, where the integrals J forH 0

and corresponding dressed integrals ~J for H are shown,respectively, in the first and the second rows.We have introduced the following notations:

p ¼ �id

dx; s1 ¼ p�2 � coth2� � �1;

s2 ¼ i�3s1;

(2.23)

P 1 ¼ �iZ� 00 Z��

� �; Z� ¼ A�

d

dxAy� ; (2.24)

S1 ¼ 0 Xy�

X� 0

!; S2 ¼ i�3S1; (2.25)

X� ¼ d

dx� ��ðxÞ; Xy

� ¼ �X��; (2.26)

Q ¼ RY� 00 RY��

� �; S ¼ RX� 0

0 RX��

� �;

(2.27)

where

��ðxÞ ¼ tanhðx� �Þ � tanhðxþ �Þ þ coth2�: (2.28)

Function ��ðxÞ, that appears in the structure of the firstorder operator X�, has the properties

��ð�xÞ ¼ ��ðxÞ; ���ðxÞ ¼ ���ðxÞ; (2.29)

and satisfies the Riccati equation of the form

�2�ðxÞ þ�0

�ðxÞ ¼ 2ðtanh2ðxþ �Þ � 1Þ þ coth22�: (2.30)

Equation (2.30) is based on the identity

5For earlier discussions of this system see [15,17,19,38].6Intertwined Hamiltonians H and ~H can be Hermitian opera-

tors of any, including matrix, nature.

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1� tanhðxþ �Þ tanhðx� �Þ þ coth2�ðtanhðx� �Þ� tanhðxþ �ÞÞ ¼ 0; (2.31)

which is the addition formula for the function tanhu.To find a map sa ! SaH , a ¼ 1, 2, the identities

A�

�d

dxþ coth2�

�¼ X��A��;�

d

dx� coth2�

�Ay� ¼ Ay��X�;

(2.32)

have been employed. Using these identities and thoseobtained from them by the change � ! ��, we find that

the first order differential operators X� and Xy� , from which

the integrals S1 and S2 are composed, are also the inter-twining operators,

X�H� ¼ H��X�; H�Xy� ¼ Xy

�H��; (2.33)

cf. (2.14) and (2.15).There are also the intertwining relations

RX� ¼ �X��R; RY� ¼ Y��R;

RZ� ¼ �Z��R;(2.34)

T X� ¼ X��T ; T Y� ¼ Y��T ; T Z� ¼ Z��T :

(2.35)

The first, X, and the second, Y, order differential opera-tors intertwine, in turn, not only the Hamiltonians H� andH��, but also the third order operators Z� and Z��,

X�Z� ¼ Z��X�; Z�X�� ¼ X��Z��;

Y�Z� ¼ Z��Y�; Z�Y�� ¼ Y��Z��:(2.36)

The operators X, Y, and Z satisfy the identities

� X��X� ¼ H� þ C22� � 1; Y��Y� ¼ H2

�;

�Z2� ¼ H2

�ðH� � 1Þ; (2.37)

X��Y� ¼ Z� þ C2�H�;

X�Z� ¼ �C2�X�H� � Y�ðH� þ C22� � 1Þ; (2.38)

Y�Z� ¼ X�H2� þ C2�Y�H�; (2.39)

where

C 2� ¼ coth2�: (2.40)

Other relations we shall need are obtained from them byHermitian conjugation, with taking into account the rela-

tions Xy� ¼ �X��, Y

y� ¼ Y��, Z

y� ¼ �Z�, as well as by the

change � ! ��.According to (2.20) with D ¼ A�, H ¼ H0, and

~H ¼ H�, the PT relations (2.37), (2.38), and (2.39) arejust the dressed free particle identities

� ðipþ C2�Þðip� C2�Þ ¼ H0 þ C22� � 1;

Ay��A�� ¼ H0; pH0p ¼ H0ðH0 � 1Þ;(2.41)

ipþ C2� ¼ ðipÞ þ ðC2�Þ;�ðip� C2�Þip ¼ ðC2�ðip� C2�Þ þ ðH0 þ C22� � 1ÞÞ;

(2.42)

H0ip ¼ ðip� C2�ÞH0 þ C2�H0: (2.43)

Return now to the information presented in Table I. Inaddition to �3, Q1, andQ2, the operators P 1, S1, S2,Q, S,R�1, T�1, andRT are identified as Hermitian integralsof motion of the superextended system H . The integrals�3, Q1, Q2, S1, S2, and P 1 are local, while the R and theintegrals which include it in their structure are nonlocal inx operators. Curiously, the nontrivial nonlocal integralQ isjust the dressed parity integral R of the free particlesystem.It is known that any n-gap quantum mechanical periodic

or nonperiodic system possesses a nontrivial Lax integralof the odd order ð2nþ 1Þ [13]. In the present case of theone-gap PT self-isospectral system H , the dressed mo-mentum operator,P 1, is (up to the numerical factor�i) thethird order Lax operator. The intertwining relations (2.36)mean that the Lax integral P 1 commutes also with the first,Sa, and the second, Qa, order integrals of the system H .The diagonal nature of the integrals (2.27) means that in

addition to the local integral Z� (Lax operator), the one-gapPT subsystem H� has also nontrivial nonlocal integrals ofmotion, RY� and RX�, see below.For the self-isospectral systemH , we have the integrals

of motion �3, R�1, T�1, and those obtained from themby a composition, R�2, T�2, RT�3, RT . The squareof each of these seven Hermitian integrals equals one, andany of them may be chosen as a grading operator �. All theintegrals from this set which include in their structurereflection operators R and T , except the integral RT ,may be obtained from the integral �3 by a unitary trans-formation,

TABLE I. Undressed (free particle), J, and dressed (PT), ~J, integrals.

1 H 0 �3 �1 �2 p s1 s2 R�1 T�1 RT R �iR�2s1

H H 2 �3H Q1 Q2 P 1 S1H S2H �R�1H T�1H �RTH Q SH

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UaðrÞ�3Uya ðrÞ ¼ r�a; where UaðrÞ ¼ 1ffiffiffi

2p ð�3 þ �arÞ;

a ¼ 1; 2; r ¼ R;T ; (2.44)

UðR;T Þ�3UyðR;T Þ ¼ RT�3;

UðR;T Þ ¼ U1ðRÞU1ðT Þ; (2.45)

UaðrÞ ¼ Uya ðrÞ; UðR;T Þ ¼ UyðR;T Þ;

U2aðrÞ ¼ U2ðR;T Þ ¼ 1:

(2.46)

There exists no unitary transformation that would relateRT with�3, or with any other integral from this set. Sinceany of the four unitary operators UaðrÞ, being composedfrom the integrals of motion, commutes withH , the latteris invariant under any of five unitary transformations gen-erated by UaðrÞ and UðR;T Þ.

With the listed above algebraic identities and intertwin-ing relations, we find that the basic local integrals of thefirst, second, and third orders, S1, Q1, and P 1, are relatedbetween themselves and the Hamiltonian H ,

S21 ¼ H þ C22� � 1; Q21 ¼ H 2;

P 21 ¼ H 2ðH � 1Þ; (2.47)

S1Q1 ¼ �i�3P 1 � C2�H ;

Q1S1 ¼ i�3P 1 � C2�H ;(2.48)

P 1S1 ¼ S1P 1 ¼ �i�3ðQ1ðH þ C22� � 1Þ þ C2�HS1Þ;(2.49)

P 1Q1 ¼ Q1P 1 ¼ i�3ðS1H 2 þ C2�HQ1Þ: (2.50)

These identities reproduce modulo H the polynomialrelations between the corresponding integrals s1, �1, and pof the free particle system H 0.

III. EIGENSTATES OF P 1, Q1, AND S1

In accordance with (2.49) and (2.50), there exists acommon basis for the integral P 1 and for one of theintegrals Q1 or S1. The two sets of corresponding eigen-states can be presented in a unified form,

�0;1�;þ ¼ �0;1ðxþ �Þ

�0;1� �0;1ðx� �Þ !

; �0;1�;� ¼ �3�

0;1�;þ;

(3.1)

H�0�;� ¼ 0; H�1

�;� ¼ �1�;�;

P 1�0;1�;� ¼ 0; � ¼ �; (3.2)

��k�;þ ¼ c�kðxþ �Þ

e�i’�ðk;�Þc�kðx� �Þ� �

; ��k�;� ¼ �3�

�k�;þ;

(3.3)

H ��k�;� ¼ ð1þ k2Þ��k

�;�;

P 1��k�;� ¼ �kð1þ k2Þ��k

�;�;(3.4)

where � ¼ Q1 or S1, and �0, �1, and c�k are thefunctions defined in (2.6) and (2.7),

�0;1Q1¼ ��0;1S1 ¼ þ1; ei’Q1

ðk;�Þ ¼ e2ik�;

ei’S1ðk;�Þ ¼ e2ik�þi�ðk;�Þ;

(3.5)

ei�ðk;�Þ ¼ e�i�ð�k;�Þ ¼ �e�i�ðk;��Þ ¼ ik� C2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ C22�

q ; (3.6)

Q1�0Q1;�

¼ 0; Q1�1Q1;�

¼ ��1Q1;�

;

Q1��kQ1;�

¼ �ð1þ k2Þ��kQ1;�

;(3.7)

S1�0S1;�

¼ �1

sinh2��0

S1;�; S1�

1S1;�

¼ �C2��1S1;�

;

S1��kS1;�

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ C22�

q��k

S1;�: (3.8)

Antidiagonal operatorsQ1 and S1 anticommute with �3,and multiplication by �3 changes their eigenstates intoeigenstates with an eigenvalue of the opposite sign. Fromthese relations we see that any pair of mutually commutingoperators, ðP 1; Q1Þ or ðP 1; S1Þ, provides the completeinformation about the Hamiltonian eigenstates.The HamiltonianH does not distinguish the eigenstates

different in index �, and does not separate the states withindex þk and �k in the continuous part of the spectrum.Lax integral P 1 distinguishes the states with indexþk and�k, but is insensitive to the index �, and does not separatethe doublet states of energy E ¼ 0 and E ¼ 1, annihilating

all the corresponding four states �0;1�;�. The integrals Q1

and S1 distinguish the states with E ¼ 0 and E ¼ 1, detecta difference between the states with � ¼ þ and � ¼ �, butdo not separate the states with indexþk and�k. The onlyintegral that detects, by its eigenvalues, a displacement 2�between the two subsystems is S1. Its eigenvalues of thebound states �0

S1;�, as well as of the continuous spectrum

eigenstates, blow up, however, in the limit of a zero shift,� ! 0.The spectrum of Q1 coincides with that of the operator

�3H . This is not casual: these two operators are theDarboux-dressed form of the free particle integrals �1

and �3, which can be related by a unitary transformation.As follows from Table II below, the nonlocal integral

R�1 commutes with both integrals Q1 and S1. Acting ontheir eingestates, it detects the nontrivial relative phasesbetween the upper and lower components of the eigenstates,

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R�1�0Q1;�

¼ ��0Q1;�

; R�1�1Q1;�

¼ ���1Q1;�

;

(3.9)

R�1�0S1;�

¼ ���0S1;�

; R�1�1S1;�

¼ ��1S1;�

;

(3.10)

R�1��k�;� ¼ ��e�i’�ðk;�Þ��k

�;�: (3.11)

This can be compared with the action of another nonlocalintegral, RT , that also changes index þk for �k of thescattering states, but does not detect the correspondingrelative phases,

RT�0�;� ¼ �0

�;�; RT�1�;� ¼ ��1

�;�;

RT��kQ1;�

¼ ���kQ1;�

; RT��kS1;�

¼ ���kS1;��:

(3.12)

The difference in the two last relations in (3.12) originatesfrom the commutativity of RT with Q1 and its anticom-mutativity with S1, see Table II.

Finally, we note that though Q1 and S1 do not (anti)commute and each of them does not distinguish indexesþk and �k of the states in the continuous part of thespectrum, according to (2.48), the Lax integral and theHamiltonian are reconstructed from them,

i

2�3½S1; Q1� ¼ P 1; � 1

2C2�fS1; Q1g ¼ H : (3.13)

Similarly, each pair of the integrals ðP 1; S1Þ or ðP 1; Q1Þallows us to reconstruct the third operator, respectively, Q1

or S1, see Eqs. (2.49) and (2.50).This information constitutes a part of a nonlinear super-

algebraic structure of the system, which we discuss in thenext section.

IV. NONLINEAR SUPERSYMMETRIES OF SELF-ISOSPECTRAL PT SYSTEM

For the grading operator � ¼ �3, the antidiagonal localintegrals Qa and Sa are identified as fermionic operators,while the diagonal integrals P 1 and P 2 ¼ �3P 1 should betreated as bosonic generators of the superalgebra. Thenonlinear superalgebraic relations (2.19) are extended

then, in correspondence with Eqs. (2.47), (2.48), (2.49), and(2.50), to the nonlinear superalgebra

fSa; Sbg ¼ 2�abðH þ C22� � 1Þ;fQa;Qbg ¼ 2�abH 2;

(4.1)

fSa;Qbg ¼ �2�abC2�H � 2�abP 1; (4.2)

½P 2; Sa� ¼ �2iððH þ C22� � 1ÞQa þ C2�HSaÞ; (4.3)

½P 2; Qa� ¼ 2iðH 2Sa þ C2�HQaÞ; (4.4)

½P 1; Sa� ¼ ½P 1; Qa� ¼ ½P 1;P a� ¼ 0; (4.5)

½�3; Qa� ¼ �2i�abQb; ½�3; Sa� ¼ �2i�abSb;

½�3;P a� ¼ 0;(4.6)

in which the Lax operator P 1 plays the role of the centralcharge.The last relation from (2.47) does not show in the (anti)

commutation relations. It displays, however, in the super-algebraic relations of the local integrals for any otherchoice of the grading operator since then at least one ofthe two integrals P a is identified as an odd, fermionicoperator. The Z2 parity � ¼ �, �A� ¼ �A, of the localand some nonlocal integrals for all the choices ofthe grading operator � is shown in Table II, whereEqs. (2.34) and (2.35) and relations

�1a bc d

� ��1 ¼ d c

b a

� �;

�2a bc d

� ��2 ¼ d �c

�b a

� �;

(4.7)

for 2� 2 matrices have been used.7

Notice that for any choice of the grading operator, thenonlocal integralQ, likeH andRT , is an even operator.

TABLE II. Z2 parity of the local and some nonlocal integrals.

� Q1 Q2 S1 S2 P 1 P 2 �3 S Q �3S �3Q RT

�3 � � � � þ þ þ þ þ þ þ þR�1 þ � þ � � þ � � þ þ � þT�1 þ � � þ þ � � þ þ � � þR�2 � þ � þ � þ � � þ þ � þT�2 � þ þ � þ � � þ þ � � þRT�3 � � þ þ � � þ � þ � þ þRT þ þ � � � � þ � þ � þ þ

7Though Hermitian unitary operators (2.44) and (2.45) alsocommute with the Hamiltonian H and their square equals one,they do not assign a definite Z2 parity to some of the nontrivialintegrals listed in Table II.

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As another example, we display the nonlinear super-algebraic relations satisfied by the local integrals for thechoice � ¼ RT ,

fSa; Sbg ¼ 2�abðH þ C22� � 1Þ; (4.8)

fP 1;P 1g ¼ fP 2;P 2g ¼ 2H 2ðH � 1Þ;fP 1;P 2g ¼ 2H 2ðH � 1Þ�3;

(4.9)

fSa;P 1g ¼ �2�abððH þ C22� � 1ÞQb þ C2�HSbÞ;fSa;P 2g ¼ 0; (4.10)

½Q1; Q2� ¼ �2iH 2�3; ½Qa;P 1� ¼ 0; (4.11)

½Qa; Sb� ¼ 2ið�abP 2 þ �abC2�H�3Þ;½Qa;P 2� ¼ �2iðH 2Sa þ C2�HQaÞ;

(4.12)

which have to be completed by Eq. (4.6).The even generator �3 appears only in (4.6) in super-

algebra with � ¼ �3, while for � ¼ RT it is presentalso in the (anti)commutation relations (4.9) and (4.12).Another, essential difference between both superalgebrasis that in the second case the constant C2� ¼ coth2� anti-commutes with the grading operator RT and has to betreated there as an odd generator of the superalgebra. Withsuch interpretation, the anticommutator (4.10) and thecommutators in (4.12) produce, respectively, even andodd combinations of the generators. In the case � ¼ �3,the C2� should be treated as the even central charge. In bothsuperalgebras, the Hamiltonian H appears as a multipli-cative central charge, which makes them nonlinear. Apicture is similar for other choices of the grading operatorshown in Table II.

The supersymmetric structure of the self-isospectralone-gap PT system generated by local integrals of motionadmits therefore different choices for the grading operator;each corresponding form of the superalgebra is centrallyextended and nonlinear. According to Eqs. (2.47) and (2.6),only the integrals P a annihilate the singlet states of theisospectral subsystemsH� andH��. On the other hand, theintegrals Sa have an empty kernel, while the Qa, a ¼ 1, 2,annihilate only the states of zero energy. Having in mindthat for any choice of the grading operator at least twointegrals from the set of the four integrals Qa and Sa areidentified as fermionic generators, we always have par-tially broken nonlinear supersymmetry, cf. this picture withthat of supersymmetry in the systems with topologicallynontrivial Bogomolny-Prasad-Sommerfield states [40].

One can find a modification of the integrals S1 and S2,which annihilate the doublet of the ground states of theself-isospectral system, by combining them with the (non-local in the shift parameter �) integralT�1. We get it usingthe explicit form (3.1) of the zero energy eigenstates of S1,

�S 1 ¼ S1 þ 1

sinh2�T�1; �S1 ¼ i�3

�S1;

�S1�0�S1;�

¼ 0:

(4.13)

Themodified integrals �Sa,a ¼ 1, 2, are odd superchargeswith respect to both choices of the grading operator,� ¼ �3

and � ¼ RT , which correspond to the discussed super-algebras. The price to pay, however, is that the integrals �Saare not only nonlocal in the shift parameter, but also are

non-Hermitian, �Sy1 ¼ S1 � sinh�12�T�1 � �S1, and simi-

larly, �Sy2 � �S2. We have used here the relation

ðsinh�12�T Þy ¼ T sinh�12� ¼ �sinh�12�T . The modi-

fied supercharges �S1 and �Sy1 satisfy, particularly, the anti-

commutation relations8

f �S1; �S1g ¼ f �Sy1 ; �Sy1 g ¼ 2H ;

f �S1; �Sy1 g ¼ 2ðH þ 2ðC22� � 1ÞÞ:(4.14)

V. HIDDEN SUPERSYMMETRY OF UNEXTENDEDONE-GAP PT SYSTEM

We show here that the unextended, one-gap reflection-less PT system is characterized by an exotic hidden super-symmetry, which can be obtained from the supersymmetryof the self-isospectral system H by a nonlocal Foldy-Wouthuysen transformation with a subsequent reduction.Indeed, the nonlocal integrals (2.27) can be obtained

from the corresponding local integrals by applying a non-local unitary transformation

~O ¼ U1ðRÞOUy1 ðRÞ; U1ðRÞ ¼ 1ffiffiffi

2p ð�3 þ �1RÞ;

Uy1 ðRÞ ¼ U1ðRÞ; U2

1ðRÞ ¼ 1; (5.1)

see Eq. (2.44). This transformation does not change the

form of the self-isospectral Hamiltonian, ~H ¼ H , while

~S1 ¼ S; ~S2 ¼ �S2; ~Q1 ¼ �3Q;

~Q2 ¼ �Q2;~P 1 ¼ iR�2P 1;

~P 2 ¼ �P 2;

gT�1 ¼ �3RT ; (5.2)

where S and Q are the nonlocal integrals (2.27). Thetransformation (5.2) diagonolizes the supercharges S1 andQ1, and may be treated as a kind of Foldy-Wouthuysentransformation. At the same time, it does not change thediagonal form of the integral P 2.

8Some similar integrals for the not self-isospectral supersym-metric PT systems were discussed in [40] in the context of shapeinvariance, see also [19]. Unlike the present case, however, theintegrals considered in [40] do not anticommute with the corre-sponding grading operator �3, and their treatment as fermionicgenerators in the superalgebraic relations is not justified there.

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The PT subsystem H�, which is just a reduction of thesystem H to the subspace �3 ¼ þ1, has therefore onelocal and two nonlocal nontrivial integrals

P 1 ¼ �iZ�; S1 ¼ RX�; Q1 ¼ RY�: (5.3)

Making use of the intertwining relations (2.34) and (2.35)

and relations (2.37) and (2.38), the Lax integral P 1 is

reconstructed from the integrals S1 and Q1,

P 1 ¼ i

2fS1; Q1g; (5.4)

cf. (3.13). Similarly, S1 (Q1) can be reconstructed from the

commutator of Q1 (S1) with P 1.Taking the integral

� ¼ RT ; (5.5)

�2 ¼ 1, ½H�; �� ¼ 0, as the grading operator, we identify

the integrals P 1 and S1 as odd, fermionic operators, while

Q1 is identified as an even, bosonic operator. They can be

supplied with two fermionic, P 2 ¼ i�P 1, S2 ¼ i�S1, and

one bosonic, Q2 ¼ �Q1, integrals,

P 2¼RT Z�; S2¼ iT X�; Q2¼ iT Y�: (5.6)

The nonlinear superalgebra generated by Sa, Qa, P a, andH� is

fSa; Sbg ¼ 2�abðH� þ C22� � 1Þ;fP a; P bg ¼ 2�abH

2�ðH� � 1Þ;

(5.7)

fS1; P 1g ¼ fS2; P 2g ¼ 0;

fS1; P 2g ¼ fS2; P 1g ¼ �2iC2�S2H�;(5.8)

½Qa; Qb� ¼ 0; ½Q1; Sa� ¼ 2iP a;

½Q2; S1� ¼ �2�C2�H�; ½Q2; S2� ¼ �2iC2�H�;

(5.9)

½Q1; P 1� ¼ �2iðH2�S1 þ C2�H�Q1Þ;

½Q1; P 2� ¼ 2�ðH2�S1 þ C2�H�Q1Þ; ½Q2; P a� ¼ 0:

(5.10)

As in the case of the superalgebra (4.8), (4.9), (4.10),(4.11), and (4.12), here the constant C2� anticommutes with

the grading operator �, and has to be treated as anodd generator of the superalgebra that guarantees, particu-larly, the correct Hermitian properties of the (anti)commu-tation relations. For instance, for the right-hand side

of the last relation in (5.8) we have ð�2iC2�S2H�Þy ¼þ2iH�S2C2� ¼ �2iC2�S2H� in correspondence with

ðfS1; P 2gÞy ¼ fS1; P 2g, where we have taken into accountthe Hermitian nature of the involved integrals, and

S2C2� ¼ �C2�S2 due to the presence of the operator Tin the structure of the supercharge S2.Notice also that the nature of superalgebra (5.7), (5.8),

(5.9), and (5.10) of the hidden supersymmetry of the PTsystem H� has differences in comparison with the bothsuperalgebras of the self-isospectral system discussed the

previous section. Particularly, the operators P a (Qa) areodd (even) generators here in comparison with the even(odd) nature of the integrals P a (Qa) in the superalgebra(4.1), (4.2), (4.3), (4.4), (4.5), and (4.6). Unlike the super-algebras (4.1), (4.2), (4.3), (4.4), (4.5), (4.6), (4.8), (4.9),

(4.10), (4.11), and (4.12), the identity Q2a ¼ H2

� does notappear in the superalgebraic relations, cf. the second rela-tion in (4.1) and the first relation in (4.11) with taking intoaccount Q2 ¼ i�3Q1.Let us look at how the basic supersymmetry generators

(5.3) act on the states of the system. Before, we note thatthough the dependence on � in the displaced Hamiltonian

H� and odd integral P 1 may be eliminated by the shift x !x� �, the parameter �2� will still be present in the

structure of the integrals S1 and Q1, as well as in thegrading operator and, in the invariant under such a shift,superalgebraic relations. Under such a shift, the reflectionR with respect to x ¼ 0, that enters into the grading

operator �, will be changed for the reflection with respectto x ¼ ��.On the other hand, though the nonshifted Hamiltonian

(2.1) is even while the Lax operator �iZ ¼ �iA ddx A

y is

odd with respect to the reflection R in the x ¼ 0 operator,

the integrals S1 and Q1 do not possess a definite paritywith respect to it.9.The eigenstates of the shifted HamiltonianH� we denote

here as in (2.6), imply that their argument is xþ �. Theeigenstates and eigenvalues of the operators (5.3) are

P 1�0 ¼ P 1�

1 ¼ 0; P 1c�k ¼ �kðk2 þ 1Þc�k;

(5.11)

Q 1�0 ¼ 0; Q1�

1 ¼ �1;

Q1ckQ1;�

¼ �ð1þ k2Þc kQ1;�

;(5.12)

S1�0 ¼ �ðsinh2�Þ�1�0; S1�

1 ¼ coth2��1;

S1ckS1;� ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ coth22�

pc k

Q1;�; (5.13)

c kQ1;�

¼ 12ðcþk � e2ik�c�kÞ;

c kS1;� ¼ 1

2ðcþk � ei’S1ðk;�Þc�kÞ;

(5.14)

9The integrals �iZ and RZ generate together with (2.2) thethird order, nonlinear superalgebra with R identified as thegrading operator, see [6].

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cf. (3.1), (3.2), (3.3), (3.4), (3.5), (3.6), (3.7), and (3.8), on

which the grading operator � acts as

��0 ¼ �0; ��1 ¼ ��1;

�ðcþk � c�kÞ ¼ �ðcþk � c�kÞ;(5.15)

�c kQ1;�

¼ �e�2ik�c kQ1;�

;

�c kS1;� ¼ �e�i’S1

ðk;�Þc kS1;�;

(5.16)

cf. (3.9), (3.10), and (3.11), where ’S1ðk; �Þ is the phase

defined by Eqs. (3.5) and (3.6).Like in the case of the extended, self-isospectral system

H , the Hamiltonian H� distinguishes here the singletstates �0 and �1, but does not distinguish the doublet

states c�k. The Lax operator P 1, instead, distinguishesthe doublet states, but does not distinguish the singlet

states: they both are its zero modes.10 The operator Q1

distinguishes all the eigenstates ofH�, but it does not detecta virtual shift � here, like the integral Q1 does not detect areal shift between the subsystems of the extended self-

isospectral system H . Only the integral S1 distinguishesall the states as well as detects a virtual shift 2� here.

Unlike the integrals P 1 and Q1, the kernel of the

integral S1 is empty: it annihilates a nonphysical state11

f�ðxÞ ¼ coshðx� �Þcoshðxþ �Þ e

x coth2�; (5.17)

in terms of which the operator (2.26) is presented as

X� ¼ f�d

dx

1

f�¼ d

dx� ðlnf�Þ0: (5.18)

The analogs of the modified supercharges (4.13) here are

�S 1 ¼ S1 þ ðsinh2�Þ�1�; �S2 ¼ i� �S1;

f�; �Sag ¼ 0; �Sa�0 ¼ 0;

(5.19)

which are non-Hermitian, odd operators. Operators �S1 and�Sy1 satisfy the aticommutation relations f �S1; �S1g ¼

f �Sy1 ; �Sy1 g ¼ 2H�, f �S1; �Sy1 g ¼ 2ðH� þ 2ðC22� � 1ÞÞ, cf. (4.14).

VI. SUPERSYMMETRY OF ONE-GAPNONPERIODIC BDG SYSTEM

The self-isospectral supersymmetric structure we havediscussed admits an interesting alternative interpretation in

terms of the associated one-gap, nonperiodic Bogoliubov-de Gennes system. In this section we reveal a set of (non)local integrals for the latter system, which generate a non-linear supersymmetric structure to be of the order eight inthe BdG Hamiltonian.Consider one of the local, first order integrals Sa as a

ð1þ 1ÞD Dirac Hamiltonian. This corresponds to theBogoliubov-de Gennes system. Depending on the physicalcontext, the function �� plays a role of an order parameter,a condensate, or a gap function [15,17–19].For the sake of definiteness, we identify S1 as a first

order Hamiltonian, HBdG ¼ S1. It is a Darboux-dressed form of the ð1þ 1ÞD Dirac Hamiltonian s1 ¼p�2 � coth2� of the free particle of mass m ¼ coth2�.The energy gap 2m ¼ 2 coth2� in the spectrum of the freeDirac particle transforms effectively by the Darboux trans-formation (2.22) into the x-dependent gap function 2��ðxÞ.The square of the free Dirac particle Hamiltonian, (2.21),which is given by the two copies of the free particle secondorder Hamiltonian, transforms into the Hamiltonian ofthe self-isospectral PT system H , whose eigenstates aregiven by Eqs. (3.1) and (3.3). Under such a transformation,the mass parameter m ¼ coth2� of the free particle systemmaps into a spatial shift 2� of the two PT subsystems,H� and H��.Operator �3 anticommutes with the BdG Hamiltonian

S1, and plays a role of the energy reflection operator. Asfollows from Table II, HBdG ¼ S1 commutes with R�1,T�2, and RT�3, any of which can be identified as thegrading operator for the BdG system. These are nonlocal inx or �, or in both of them, trivial integrals of HBdG. Anontrivial local BdG integral is P 1. The BdG Hamiltoniananticommutes with the nonlocal integral S of the self-isospectral PT system. The latter is just the Foldy-Wouthuysen transformed, diagonal form of the BdGHamiltonian S1, being a Darboux-dressed form of theoperator

� iR�2s1 ¼ R�� d

dxþ �3 coth2�

�; (6.1)

see Table I. Operator (6.1) is the Foldy-Wouthuysen trans-formed, diagonal form of the free Dirac particleHamiltonian s1. The operator

�3S ¼ ðR�1ÞS1 (6.2)

is then a nonlocal integral of HBdG, ½S1; �3S� ¼ 0. HBdG

still has one more, nontrivial nonlocal integral. To identifyit, we note that with respect toR�1, T�2, orRT�3, thelocal integral P 1 is identified, respectively, as the odd,even, or odd operator, see Table II, while the nonlocalintegral �3S has, respectively, even, odd, and, once again,odd Z2 parities. This means that in dependence on thechoice of the grading operator, we have to calculate eitherthe commutator or anticommutator of these two integrals.We find

10The third state annihilated by the third order differentialoperator Z� is coshðxþ �Þ, which is a nonphysical state of afree particle Hamiltonian (2.3) of zero eigenvalue [41].11The state (5.17) is a nonphysical eigenstate of H� of eigen-value �sinh�22�. The second state annihilated by Q1 isðsinh2xþ 2x cosh2�Þ= coshðxþ �Þ, which is a nonphysical ei-genstate of H� of eigenvalue 0.

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fP 1; �3Sg ¼ 0; ½P 1; �3S� ¼ �2iF ; (6.3)

where

F ¼ �i�3SP 1 ¼ C2�SðS21 � C22� þ 1Þ þ �3QS21 (6.4)

is the third basic, nontrivial BdG nonlocal integral, whichis a Darboux-dressed integral Rð ddx � C2�Þ d

dx of the free

Dirac particle. The Z2 parities of F with respect to R�1,T�2, or RT�3 are, respectively, �, �, or þ, where theanticommutativity of C2� with T�2 and RT�3 has to betaken into account.

Summarizing, for each of the three possible identifica-tions of the grading operator for the BdG system, R�1,T�2, or RT�3, one of the basic integrals, respectively,�3S, P 1, or F , is identified as the even generator, whilethe two other integrals are identified each time as theZ2-odd supercharges, see Table III.

The set of the (anti)commutation relations (6.3) has to beextended then by

f�3S;F g ¼ 0; ½�3S;F � ¼ 2iP 1S21; (6.5)

fP 1;F g ¼ 0;

½P 1;F � ¼ 2iðS21 � C22� þ 1Þ2ðS21 � C22�Þ�3S;(6.6)

P 21 ¼ ðS21 � C22� þ 1Þ2ðS21 � C22�Þ; ð�3SÞ2 ¼ S21;

(6.7)

F 2 ¼ S21ðS21 � C22�ÞðS21 � C22� þ 1Þ2: (6.8)

The action of the Lax integralP 1 on the eigenstates of theHamiltonian HBdG ¼ S1 is given by Eqs. (3.2) and (3.4),while the action of the BdG integrals �3S and F can beeasily found by making use of Eqs. (6.2), (6.4), (3.1), (3.2),(3.3), (3.4), (3.5), (3.6), (3.7), (3.8), (3.10), and (3.11).

The spectrum of the HBdG ¼ S1 is symmetric,ð�1;�E1Þ [ �E0 [ E0 [ ðE1;þ1Þ, where E0 ¼sinh�12�, E1 ¼ coth2�. The eigenvalues�E0 of the boundstates, and the eigenvalues �E1 of the edge states of thecontinuous parts of the spectrum are nondegenerate. Thecontinuous bands are separated by the gap 2E1 ¼ 2 coth2�,while E2

1 � E20 ¼ 1. All the corresponding singlet states are

annihilated by the integrals P 1 and F , while �3S�0S1;�

¼�E0�

0S1;�

, �3S�1S1;�

¼ E1�1S1;�

, cf. Eq. (3.8). The eigen-

states and eigenvalues of �3S and F in the doubly degen-erate continuous parts of the spectrum are given by

�3Sð�þkS1;�

� ei’S1ðk;�Þ��k

S1;�Þ

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ E2

1

qð�þk

S1;�� ei’S1

ðk;�Þ��kS1;�

Þ; (6.9)

F ð�þkS1;�

� iei’S1ðk;�Þ��k

S1;�Þ

¼ �kð1þ k2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ E2

1

qð�þk

S1;�� iei’S1

ðk;�Þ��kS1;�

Þ:(6.10)

Since all the three basic integrals P 1, �3S, and F com-mute with �3, only the BdG Hamiltonian S1 detects adifference between the states with opposite values of thelow index �, see Eq. (3.8).Let us discuss the structure of the superalgebra of the

BdG system. The trivial integralsR�1, T�2, andRT�3

generate between themselves the three-dimensionalClifford algebra, i.e., the same algebra that the �i, i ¼ 1,2, 3, do. For any choice of the grading operator, twodifferent basic odd supercharges anticommute. The squareof the each basic integral, �3S, P 1, andF , is a polynomialin H2

BdG ¼ S21 of the order, respectively, 1, 3, and 4. A

commutator of any two basic integrals produces, modulo acertain polynomial of H2

BdG ¼ S21, a third integral. As a

result, for any choice of the grading operator, the super-algebra has a somewhat similar structure to be a nonlinearsuperalgebra, in which the H2

BdG ¼ S21 plays a role of the

multiplicative central charge.As an explicit example, consider the case with � ¼

R�1 chosen as the grading operator, and denote

A 1 ¼ P 1; A2 ¼ i�A1; F 1 ¼ F 2;

F 2 ¼ i�F 1; B ¼ �3S;(6.11)

where Aa and F a, a ¼ 1, 2, are identified as the oddgenerators, while B is the even generator. In these nota-tions, a nonlinear superalgebra of the one-gap, nonperiodicBdG system can be presented in a compact form,

fAa;Abg ¼ 2�abðS21 � C22�ÞðS21 � C22� þ 1Þ;fAa;F bg ¼ 0;

(6.12)

fF a;F bg ¼ 2�abS21ðS21 � C22�ÞðS21 � C22� þ 1Þ2; (6.13)

½B;Aa� ¼ 2iF a; ½B;F a� ¼ 2iS21Aa: (6.14)

This is a nonlinear superalgebra of the order eight in theBdG Hamiltonian HBdG ¼ S1.

VII. DISCUSSION AND OUTLOOK

Our analysis of nonlinear supersymmetry of the one-gap, reflectionless self-isospectral Poschl-Teller systemwas based on a mirror symmetry and a related Darbouxdressing.

TABLE III. Possible grading operators and Z2 parities of thebasic BdG integrals.

� P 1 �3S F

R�1 � þ �T�2 þ � �RT�3 � � þ

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Mirror symmetry has a twofold nature here. On the onehand, it is generated by a spatial reflection, and by areflection of the parameter of a shift of the two PT sub-systems. On the other hand, in the Darboux-Crum mapbetween the two PT subsystems, a free particle systemappears as a virtual mirror, by means of which the secondorder Darboux-Crum transformation between the mutuallyshifted PT subsystems factorizes into a sequence of the twofirst order Darboux transformations.

In this construction, all the trivial and nontrivial gener-ators of the supersymmetry of the self-isospectral PT sys-tem appear as the Darboux-dressed integrals of the freespin-1=2 particle system described by the second orderHamiltonian. The first order, one-gap Bogoliubov-deGennes system associated with the self-isospectral secondorder PT system is just a dressed free massive Diracparticle. In such a picture, a mass parameter of the freeDirac particle transforms effectively into a gap function ofthe BdG system. The Dirac mass maps into the parameterof the mutual shift (displacement) of the two subsystemsfor the second order self-isospectral PT system.

The key role in the exotic nonlinear supersymmetry ofthe one-gap, reflectionless self-isospectral Poschl-Tellersystem and the associated first order Bogoliubov-deGennes system is played by the third order Lax operator,which is a diagonal integral for the both systems, and is adressed momentum operator of the corresponding freeparticle systems. In dependence on the choice of the grad-ing operator, for which there are, respectively, seven (PT)

and three (BdG) possibilities, it plays the role of one of theeven or odd integrals of the motion. Supersymmetric struc-tures of both, PT and BdG, systems also include two morebasic nontrivial integrals, which provide a factorization ofthe Lax operator, modulo a corresponding second or firstorder Hamiltonian.The analysis, based on the mirror symmetry, may be

extended directly for the n-gap nonperiodic case by anappropriate generalization of the Darboux-Crum transfor-mation. Our approach may also be applied to the case ofn-gap periodic, second order Lame quantum systems, andto the associated periodic BdG systems. Since the one-gapPoschl-Teller potential may be achieved as a limit case ofthe Lame one with n ¼ 1, this, particularly, will allow us toanalyze in a new light a connection between the algebraicstructure associated with the previously observed hiddensupersymmetry in Lame systems [31,32] with the corre-sponding structure studied here. All these generalizationswill be presented elsewhere.

ACKNOWLEDGMENTS

The work of M. S. P. has been partially supported byFONDECYTGrant No. 1095027, Chile and by the SpanishMinisterio de Educacion under Project SAB2009-0181(sabbatical grant). L.-M.N. has been partially supportedby the Spanish Ministerio de Ciencia e Innovacion (ProjectMTM2009-10751) and Junta de Castilla y Leon(Excellence Project GR224).

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