On the absolutely continuous spectrum of block operator matrices

Albeverio, Sergio; Konstantinov, A. Yu.

doi:10.1002/mana.200510661

Cited by 1 publication

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“…Theorem 6 is an extension of a theorem by Denisov [10, Theorem 1.2]. A proof can also be found in Albeverio and Konstantinov [1]. We give a proof in Appendix C following arguments given in [2,10].…”

Section: Proof Of Theoremmentioning

confidence: 77%

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

Froese

Hasler

Spitzer

2010

Math Phys Anal Geom

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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...

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Section: Proof Of Theoremmentioning

confidence: 77%

“…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 operator (1) H := ∆ + D + q , which is defined as…”

Section: Model and Statement Of Resultsmentioning

confidence: 99%