Physica 140A (1986) 70-77 North-Holland, Amsterdam

SOME RIGOROUS RESULTS FOR RANDOM AND QUASI-PERIODIC POTENTIALS

Thomas SPENCER School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA

We review some mathematical results and conjectures for random and quasi-periodic SchrSdinger operators. The role of localization in certain nonlinear systems is also discussed. In addition we present some general arguments which show that the correlation length exponent v >1 2/d for a wide class of d dimensional random systems including the dilute Ising ferro-magnet.

1. Introduction

This talk will focus on some recent mathematical results and conjectures for the SchrSdinger operator H = -e2A + o, defined on the lattice Z d or the continuum R d. The potential will be denoted by v ( x ) or o ( j ) for x e R d and

j ~ Z d and the lattice laplacian is defined by

A f ( j ) = ~_~ ( f ( i ) - f ( j ) ) . i:li-jl=l

The constant e will be specified later. We shall be primarily interested in the behavior of solutions to the equation

H + = E+. The analysis of this equation for the class of potentials considered below generally involves overcoming small divisor problems. Roughly speaking this is achieved by a multi-scale analysis related to KAM (Kolmogorov, Arnold, Moser) methods and to renormalization group techniques.

Four classes of potentials will be considered corresponding to different physi-

cal situations. 1) Periodic potentials. Here the hamiltonian describes, for example, the motion

of non-interacting electrons in a perfect crystal at zero temperature. 2) Random potentials on Z d. We assume that v ( j ) are independent random

variables with a common bounded distribution density g (v)do . For example if g is uniformly distributed over the interval [ - w, w] then H is called the Anderson tight binding hamiltonian which he introduced and analysed to study electrons in

crystals with impuritiesl).

0378-4371/86/$03.50 © Elsevier Science Publishers B.V. (North-Hol land Physics Publishing Division)

RANDOM AND QUASI-PERIODIC POTENTIALS 71

3) Quasi-periodic potentials on Z 1 or R 1. We restrict ourselves to the case where there is just one incommensurate frequency a present and the potential has the form:

a) v ( x ) = c o s ( x ) + c o s ( a x + O ) , x , O ~ R , b) v ( j ) = cos2~r(ctj + 0), j ~ Z.

The number a is assumed to be poorly approximated by rationals. More precisely we assume

Isin nr:a[ >1 Co/n 2. (1)

The set of irrationals satisfying (1) for some C O has measure one. Although we shall state our results for the cosine, this function can be replaced by any smooth, even, periodic function with precisely two non-degenerate critical points.

Quasi-periodic potentials naturally arise if one linearizes about a quasi-peri- odic orbit. See 4) below. They also arise in the study of a quantum electron moving in a perfect crystal subjected to a periodic potential whose period is incommensurate with that of the lattice.

4) Nonlinear systems. Let us consider for example the discrete time pendulum or standard map defined by the recursion relation

e 2 ( A x ) j =- e2(xj+, + xj_ 1 - 2x j ) = sin xj. (2)

This naturally defines an area preserving transformation of the torus to itself. Note that xj depends on the initial data (Xo, Xl) hence so does the potential defined by v ( j ) = cos (xj), j ~ Z. If we wish to determine the sensitivity of the orbit xj on x o we differentiate (2) with respect to x o and obtain the equation

HLk = 0, where ~ ( j ) = Oxi/Ox o.

In this case we see that the Schrrdinger equation arises from linearizing about a nonlinear orbit. Most of our comments on (2) will be conjectural.

We shall also present some recent results on an infinite chain of nonlinearly coupled oscillators whose hamiltonian has the form

~,p2 + w2qi 2 + e f (q i _ qi+l), (3) i

where the w i are assumed to be independent random variables and f is an analytic function which satisfies f ( q ) = O(q 4) for small q.

72 T. SPENCER

2. Results and conjectures

1) When e > 0 it is well known that the spectrum for H in the periodic case is purely absolutely continuous. Moreover all generalized eigenfunctions (i.e. poly- nomially bounded solutions to HqJ = E+) are bloch waves

qJ ( x ) = p ( x ) exp ( ikx) ,

where p is a periodic function having the same period as v. 2) For the case v ( j ) is random, let us first note that the spectrum of H when

e = 0 is an interval of dense pure point spectrum with probability one. In fact the eigenvalues are v ( j ) and the eigenfunctions are delta functions on the lattice. Gold'sheid, Molchanov and Pastur 2) were the first to prove that a similar picture holds when d = 1 for all e. More precisely they proved that there is a basis of exponentially localized eigenstates, i.e. localization. Physically localization im- plies that under time evolution the electron's wave packet does not spread and the electron is trapped. Thus there is no conductivity or diffusion in the system. Recently Carmona, Klein and Martinelli (preprint), have extended these results to the case where the distribution is singular, e.g. oj = + 1.

When d >t 2 it is known that there is always an interval (near the edge of the spectrum), consisting only of localized states, provided that the density g is bounded. Furthermore if e is small, then all states are localized. These results were independently ob ta ined 'by a number of authors, Simon and Wolff3), Delyon, Levy and Souillard4), and FriShlich, Martinelli, Scoppola and SpencerS). When d = 2 all states are believed to be localized for all e, but there are no rigorous results of this kind. When d >i 3 there should be a band of absolutely continuous spectrum [E m, EL] corresponding to extended states provided e is large. However the only results of this kind are known when the lattice Z d is replaced by the Bethe lattice6). Er~ and E " are called mobility edges, since electrons should be mobile in this energy range and conduction should occur.

The basic estimate needed to establish the above results on localization is the exponential decay of the Green's function or equivalently when d = 1 the positivity of the Liapunov exponent. Let E be fixed. In one dimension we say that the Liapunov exponent , / (E) is positive if with probability one

( H - E ) ¢ ( j ) = 0

only has solutions which grow at an exponential rate ~,(E) as j goes to + or - infinity. Equivalently the Green's function satisfies

RANDOM AND QUASI-PERIODIC POTENTIALS 73

[G(E,0 , j ) [ - - - [ (E - H ) - I ( 0 , j ) ] ~< Cvex p - "r(E)lJl. (4)

where Co is a v dependent constant which is finite with probability one. In one dimension the positivity of y is due to FurstenbergT), see also BorlandS). Recently Kotani 9) showed that y (E) > 0 for almost all E for a wide class of "non-deterministic" potentials.

For d >/2 the decay of G when either e is small or E lies in the band tail is due to Fr~Shlich and Spencer/°). We refer to refs. 11 and 12 for mathematical reviews.

3) Quasi-periodic case. Some time ago Dinaburg and Sinai 13) proved that in the continuum there are always high energy Bloch type eigenfunctions:

~ ( x ) = q(x) exp ( - i k ( E ) x )

via KAM methods. Here q is a quasi-periodic function of x. On the lattice, in special case v ( j ) = c o s ( a j + 0), Aubry 14) and Herma~ 5) have very elegant proofs of the positivity of v (E) for all E when e 2 < ½ and Bellissard et al. 16) showed that for small e there are exponentially localized eigenstates. The following results are due to Fr/Shlich, Wittwer and Spencer 17) and hold for a set of 6 of measure 1.

For case 3a) there are an infinite set of low energy eigenstates which decay exponentially fast provided e is small.

For case 3b). If e is small there are only localized states. In other words exponentially decaying eigenstates form a basis. Furthermore, these eigenstates have 2 n peaks where n = 0, 1, 2 . . . . which are self-similar under reflection. For example if ~k is a wave function with two peaks at 0 and l, then there is a constant C such that for [j[ ~< l/3,

C~b(j) = ~ b ( l - j ) + O(exp - 1). (5)

If the wave function ~b has four peaks at 0 < l I < l 2 - - l ' < 12, then l ' = l 1 and l 2 >~ exp cl 1. Moreover

Iv (0) - v (l 2)1 ~< exp ( - const. 12)

which by (1) and the eveness of cosine implies

I v ( j ) - v(12 - J ) l ~< exp - cl2.

This relation implies (5) with l replaced by l 2.

74 T. SPENCER

Remarks. Sinai has also proved localization for case 3b. We believe that the methods for the lattice can be extended to prove that for the continuum (case a):

i) If e is large there are only Bloch type eigenstates. ii) If ~ is small there are only localized states at low energy. These states

exhibit a self-similar structure as described for case b).

4) An outstanding problem for standard map is to prove that the Liapunov exponent ~/(E = 0) > 0, i.e. ~k(J) = OxffOXo grows exponentially fast in j , for a set of (x 0, Xx) of positive measure. Equivalently the Green's function H- I (0 , j ) decays exponentially fast. This means that there is sensitive dependence on initial data and the theory of Pesin assures us that there is an ergodic component of positive measure on which the orbit moves "stochastically". Unfortunately the only results of this kind are known for certain piecewise smooth nonlinearities as in the case where the sine in (2) is replaced by a sawtooth function. We also expect that the spectrum of H consists of dense point spec t rum- (localized states) for almost all (x0, xx) provided e belongs to a Cantor set of positive measure and I~1 << 1, Hence for e in the Cantor set we expect v ( j ) = cos(x j ) to behave as in the random case. We remark that the Green's function G also has small divisors E, -1, since E = 0 is in the spectrum of H. This reflects the nonuniform hyperbolicity of the dynamical system (2). However these small divisors are of an entirely different kind from those encountered in conventional K A M situations where one is searching for integrable or quasi-periodic motions.

Next we briefly describe some results for the infinite chain of oscillators (3) done in collaboration with FrShlich and Wayne18). We show that if e is small, there are infinite dimensional, invariant tori of spacially localized, time almost periodic solutions to the equations of motion with high probability. In particular for some fixed i, [p~(t) + q~(t)l >/Const. > 0 and the local energy does not go to zero. If the wj are all equal, then typically there are theorems which show that the local energy goes to zero since the wave packet can spread. The existence of these localized waves follows by using a variant of KAM methods. Much work remains to be done in understanding how general solutions to the equations of motion behave. Also our methods cannot yet handle nonlinearities of the form f (q) = d~(q 2) for small q unless f(q) = q2 in which case the problem is linear and equivalent to a random SchrSdinger equation. See ref. 19 for related results.

3. Ideas on critical exponents

We now turn to a discussion of the correlation length ~ and the associated critical exponent v for a class of random systems which include the dilute Ising

RANDOM AND QUASI-PERIODIC POTENTIALS 75

model, percolation and the random potential. In the case of random potentials, ~(E) -1 is the exponential rate of decay of the Green's function G ( E ) for E < E m where E m is the mobility edge. Let A be a cube centered at the origin of width 2L and let G A be the Green's function of H restricted to A with Dirichlet boundary conditions. We define XL(E, o) to be the characteristic function of the event

o: max E cA(e,y,x) >1}. JYl <~L/2 x~OA

Then

eL(E) = fx(e , v) H g(o,) do, i ~ A

is the probability that the event occurs and measures the sensitivity of G A near the origin to the boundary conditions.

Theorem. There is a c o > 0 such that if PL(E) <<, c o for some L, then the Green's function G (E) decays exponentially fast with probability one and the correlation length ~(E) ~< Const. L ( E ) where P~.tE)(E) --- c o. See refs. 10 and 11 for details.

The same result holds for the dilute Ising ferromagnet with E m replaced by fie (the inverse critical temperature) and IG(E, y, x)l is replaced by (o e, ox)(fl ).

We now describe work done in collaboration with J. and L. Chayes and D. Fisher2°). It suggests that u > / 2 / d for a wide class of random systems. J. and L. Chayes have already proven this fact for percolation. There are three basic ingredients for the argument:

i) PL(E) = PI.(E + eL -d/2) + O(e).

Proof Since x ( E + 8, o + 8) = x ( E , v) and X ~< 1 we have

P (E + fx(E, ) H g(v, + 8)dr i

and

8 d

76 T. S P E N C E R

By the Schwarz inequality,

g'(v, + ,H 1,e (e + e (e)l .< fo d f E g(v i + s) g(v i + s)dvi

~ ~, k~Ag(Vp)dvp

< ~IAI t/2 (g ' (O0) 2 J g(vo) dvo ~< 8Ld/2const.

In the last line we used the fact that i # j terms give 0 contribution, i) now follows easily for a wide class of smooth g.

ii) There is a c 1 >/c o such that

P L ( E m ) ~ c 1 f o r a l l L .

This result is an easy consequence of the theorem, for if PL(Em) < C o for some L then G(E m + e) for some small e decays exponentially. This contradicts the definition of the mobility edge.

iii) cons t .L (E) / log L ( E ) <~ ~(E) <~ const .L(E). The upper bound follows from the theorem. The lower bound can also be established for percolation and the dilute Ising ferromagnet. It has not yet been established for random poten- tials. However L ( E ) is often used as a working definition of the correlation length.

To prove that v >~ 2 / d choose c o ~< l c I and note that

PL(E)( Em) -- PL(E)( E ) >~ C 1 -- C O ~ C O > O.

Hence by i)

IEm - E I >1 e L ( E ) -d/2

for some small e depending only on c 0. Clearly by iii) ~(E) > / (E m - E ) -2/d, modulo logarithmic corrections. Analogous arguments hold for dilute ferromag- netic Ising models and for percolation.

Remark. There is as yet no satisfactory mean field theory for localization and the upper critical dimension still remains a topic of some controversy.

RANDOM AND QUASI-PERIODIC POTENTIALS 77

References

1) P. Anderson, Phys. Rev. 109 (1958) 1492. 2) I. Gold'sheid, S. Molchanov and L. Pastur, Funct. Anal. App. 11 (1977) 1. 3) B. Simon and T. Wolff, Comm. Pure App. Math. 39 (1986) 75. 4) F. Delyon, Y. Levy and B. Souillard, Commun. Math. Phys. 100 (1985) 463. 5) J. FriShlich, F. MartineUi, E. Scoppola and T. Spencer, Commun. Math. Phys. 101 (1985) 21. 6) H. Kunz and B. Souillard, J. Physique Lett. 44 (1983) L411. 7) H. Furstenberg, Trans. Amer. Math. Soc. 108 (1963) 377. 8) R. Bodand, Proc. Roy. Soc. A274 (1963) 529. 9) S. Kotani, Proc. Toniguchi Conf., Katata, 1982 and Proc. AMS Conf. on Random Matrices, J.

Cohen, ed., 1985. 10) J. FrShlich and T. Spencer, Commun. Math. Phys. 88 (1983) 151. 11) T. Spencer, in Proc. Les Houches Summer School 1984, K. Osterwalder and R. Stora, eds., to

appear. 12) R. Carmona, Ecole d'Et6 de Probabilities XIV, Saint Flour, 1984. Lecture notes in Mathematics

1180. 13) E. Dinaburg and Ya Sinai, Funct. Analysis and App. 9 (1975) 279. 14) S. Aubry, Solid State Sci. 8 (1978) 264. 15) M. Herman, Comment Math. Heivetici 58 (1983) 453. 16) J. Bellissard, R. Lima and D. Testard, Commun. Math. Phys. 88 (1983) 207. 17) J. FrShlich, T. Spencer and P. Wittwer, to appear. 18) J. Frthlich, T. Spencer and C. Wayne, J. Stat. Phys. 42 (1986) 247. 19) M. Vittot and J. BeUissard, to appear. 20) J. Chayes, L. Chayes, D. Fisher and T. Spencer, to appear.