On the AC Spectrum of One-dimensional Random Schrödinger Operators with Matrix-valued Potentials

Abstract

ABSTRACT. We consider discrete one-dimensional random Schrödinger operators with decaying matrix-valued, independent potentials. We show that if the ℓ 2 -norm of this potential has finite expectation value with respect to the product measure then almost surely the Schrödinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schrödinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon, Simon, and Souillard [8]. MODEL AND STATEMENT OF RESULTSIn this paper we are interested in the absolutely continuous (ac) spectrum of quasi one-dimensional random Schrödinger operators with decaying potentials. To this end, it is convenient to formulate the problem in terms of matrix-valued potentials on the one-dimensional lattice, Z.Let us first introduce some standard notation that is used throughout this paper. If H is an operator on some Hilbert space H , then we denote by ρ(H), σ(H), σ ac (H), σ ess (H) its resolvent set, spectrum, ac spectrum, respectively its essential spectrum. By H we denote the operator norm of H.For some m ∈ N, let Sym(m) denote the set of real symmetric m × m matrices. Let D ∈ Sym(m) be some fixed matrix and let q = (q n ) n∈Z be a family of independent Sym(m)-valued random variables. We assume here that (i) the mean of each random variable q n is zero and (ii) there is a compact set K ⊂ Sym(m) so that the support of each q n is contained in K. By ν n we denote the probability measure of q n . The probability measure for q is then the product measure ν = ⊗ n∈Z ν n . We use the notation E to denote the expectation value with respect to this product measure, ν.On the Hilbert space ℓ 2 (Z; C m ) (of C m -valued functions on Z equipped with the usual Euclidean norm) we consider the operatorwhich is defined asTo state the first result of this paper we introduce the following set which depends on the (eigenvalues of the) constant Note that ∆ + D is equivalent to the (nearest neighbor) Dirichlet Laplace operator on ℓ 2 (Z × C ). The eigenvalues of D are indexed by n ∈ C and are given byIf d ≥ 2 this set is empty unless L = 1. If d = 1, then I D is non-empty but its length converges to 0 as L tends toRemarks: On the full two-dimensional lattice Z 2 , Bourgain [3] proved σ ac (∆+q) ⊇ σ(∆) for Bernoulli and Gaussian distributed, independent random potentials whose variances decay faster than |n| −1/2 . In [4], Bourgain improves this result to the weaker |n| −1/3 decay rate. For a deterministic potential, q, onIn dimension one this was proved by Deift and Killip [7].A recent improvement of this result has been obtained by Denisov [11]. In the analogous continuous setting, progress has been made towards this L 2 -conjecture e.g. by Denisov [9] and Laptev, Naboko, and Safronov [19]. For additional references see [5], [24]. PROOFS OF THEOREM 1 AND 2In order to prove the two main theorems in this paper we will study the Green's functions defined byHere, P n denotes the orthogonal projections of H = ℓ 2 (Z; C m ) onto the s...

Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph

“…In our example these would be the energies contained in the intervals [−2 √ 2, −2 √ 2 + 1] and [2 √ 2 − 1, 2 √ 2]. The analogous problem for slowly decaying random potentials on the strip was considered in [FHS3], but the methods used there do not apply to the Anderson model.…”

Transfer matrices for discrete Hermitian operators and absolutely continuous spectrum

“…Another special case of Theorem 5 gives a result on random decaying matrix potentials on a strip. For uniform compactly supported matrix potentials this result has been already proved in [FHS3]. Therefore, let G 3 = N 0 ×S be the product graph of a finite graph S and the half-line N 0 , hence 2 (G 3 ) = 2 (N 0 ) ⊗ 2 (S) = 2 (N 0 , C s ) and we choose S n = {n} × S and may use S = {1, .…”

“…This is already a non-one-dimensional problem and one has some (random) stair-like graph. As a by-product we also obtain the result for random decaying potentials on the strip as in [FHS3] (cf. Theorem 7).…”

A Geometric Approach to Absolutely Continuous Spectrum for Discrete Schrödinger Operators

“…(ii) In [12], we have extended Theorem 3.5 to matrix-valued potentials, and applied to (random) Schrödinger operators on a strip. By Proposition 3.2 there is a disk B ⊂ H so that z 0,n := Φ 0 • · · · • Φ n (z λ ) ∈ B for all n ≥ 2 and potentials q with values in a compact set K. Moreover, by Theorem 3.3, G λ (0, 0) = lim n→∞ z 0,n .…”

“…We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schrödinger operators developed in [9][10][11][12]. We study decaying potentials in one dimension and present a simplified proof of ac spectrum of the Anderson model on trees.…”