1 Centre de Physique Theorique, Ecole Poly technique, F-91128 Palaiseau, ... of Physics, Mathematics and Astronomy, California Institute of Technology,.
Communications in Commun. Math. Phys. 109, 157-165 (1987)
Mathematical
Physics
© Springer-Verlag 1987
Localization for Off-Diagonal Disorder and for Continuous Schrδdinger Operators1 Franςois Delyon1, Barry Simon2'**, and Bernard Souillard1 1
Centre de Physique Theorique, Ecole Poly technique, F-91128 Palaiseau, France Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA 2
Abstract. We extend the proof of localization by Delyon, Levy, and Souillard to accommodate the Anderson model with off-diagonal disorder and the continuous Schrόdinger equation with a random potential.
1. Introduction
New proofs of localization were recently found by Delyon et al. [1-3], Simon et al. [4], and Simon and Wolff [5]. These proofs arose in an effort to understand some very interesting work of Kotani [6] on the sensitivity of the nature of the spectrum with respect to boundary conditions for one-dimensional systems on a half-line, a work itself connected with a work of Carmona [7]. These proofs work for the Anderson model with diagonal disorder, which is the Hamiltonian H on l2(Zd) given by (Hu)(x)= Σ u(y) + bnu(x). (1) The results hold for one-, quasi-one, and multi-dimensional systems in appropriate domains of the parameters and for large classes of (possibly correlated) random processes for the Vs. In this note we will study the case of off-diagonal disorder for discrete equations and also the case of the continuous Schrόdinger equation with a random potential. It is already known in one dimension in some situations (for example assuming independence of the random parameters and some regularity of the distribution of the random variables) that a discrete Schrδdinger equation with off-diagonal disorder has only pure point spectrum and exponentially decaying eigenfunctions [8]. The same result is also known for the continuous Schrodinger operator, under some conditions of independence of the potential and regularity of its distributions [9-12]. For a large bibliography on the topic of random discrete * Work supported in part by Nato under Nato Grant Φ 346/84 ** Research partially supported by USNSF under grant DMS 84-16049
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and continuous Schrodinger equations, we refer to [13] and for a review of the mathematical results in this field we refer to [14]. In the present paper, we give a new proof of these results, based on the proof of [1-3] which is much simpler than the previous ones and in addition applies to many new situations. The preprint [15] presents a distinct but related proof of Theorem Γ below. The approach of the present paper is basically the same as the one of [1-3] for the Anderson model (1) and may be summarized as follows: we assume that the realization of the random parameters is such that for one-dimensional systems the Lyapunov exponent associated to the equation Hu = Eu is strictly positive for almost every E (alternatively for higher dimensions, we assume that the realization of the random parameters is such that for almost every E the Green's function decays exponentially with the distance). These hypothesis imply that for a.e. E any solution of Hu = Eu (H being possibly modified by a local perturbation) either decays or increases exponentially. Then we get this property also for spectrally almost every E, at least for almost every perturbation of H in some appropriate class. Since generalized eigenfunctions are known to increase at most polynomially, it follows that this set of operators H has only pure point spectrum and exponentially decaying eigenfunctions. It then remains to check if the previous set of H has full measure when the parameters are chosen according to some given random process. In Sect. 2 below, we give the basic results for discrete and continuous onedimensional systems. Then in Sect. 3 we give with some details the proof for the case of the one-dimensional discrete model with off-diagonal disorder and in Sect. 4 the proof for the continuous one-dimensional Schrodinger equation. Finally in Sect. 5 we apply the results of Sects. 2-4 to various specific random distributions of the parameters, state some extensions and present the results for multi-dimensional systems together with the applications. 2. Definitions and Results for the Discrete and Continuous One-Dimensional Cases Let us first introduce the discrete one-dimensional model with off-diagonal disorder: It is the self-adjoint operator, acting on L2(Z), defined as (Hfi)(n) = αn+1φ + l) + α n φ-l),
(2)
and we suppose for simplicity the an to be strictly positive. If u is a solution of the eigenvalue equation Hu = Eu, then its components satisfy an equation of the form
where Mn is a two by two matrix called the transfer matrix which depends on αn, an+ί and E. Let B be a fixed interval of 1R and denote by L the normalised Lebesgue measure on B. We assume the following hypothesis: Hypotheseίs HI. The αw's (for n> 1) are such that for L a.e. E, there exists a vector v = (u(l), u(2)) satisfying: limsupn~ 1 log||M n M n _ 1 ...M 2 ι;||2
Assuming Lemma 2, we get that for a.e. (aθ9aί9a2\ for |μί>ί+ι(dE)| a.e. E in B (0^i^2) any generalized eigenfunction of H decays exponentially at +00.
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Consequently this conclusion holds for μiΛ(dE) + μ2ί2(dE) a.e. E, and hence for spectrally a.e. E. This ends the proof of Theorem 1 assuming Lemma 2. Π We now turn to the Proof of Lemma 2. The proof relies on the following lemma that we will prove later: Lemma 3. The average of \μ0ti(dE)\ with respect to a0 is uniformly continuous with respect to the Lebesgue measure, that is ldaQ\μQΛ(dE)\£C
absolutely (7)
dE.
Indeed, assuming HI and Lemma 3 we have that for a.e. α0 then for |μ0 ±(dE)\a.e. E in B, the inequality (3) of the hypothesis HI is satisfied. Since det(MπMπ_ ! . . .M2) = ajan is bounded from above and below, for |μ0> ^dE^-a.e. E on B any solution of Hu = Eu decreases or increases exponentially at +00. Furthermore, any generalized eigenfunction is polynomially bounded and thus has to decrease exponentially at +00 for |μ0> 1(dE)|-a.e. E on B. This yields Lemma 2 assuming Lemma 3. Π We now prove the Lemma 3 : Proof of Lemma 3. Let HL be the restriction of H to /2([ - L, L]) and let |μ£, ^(dE)\ denote the absolute value of its spectral measure; thus |μo, ι(^)l =
Σ
l£j^2L+l
|f*/0)t
