Spectra of Anderson type models with decaying randomness

Krishna, M.; Sinha, Kalyan B.

doi:10.1007/bf02829590

Cited by 6 publications

(4 citation statements)

References 13 publications

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“…A deterministic version of a result in the direction of Theorem 2 has been announced by Molchanov and Vainberg [20] but, to the best of our knowledge, has not yet been published. Other previous work by Kirsch, Krishna, Obermeit and Sinha on decaying potentials can be found in [17], [14] and [18]. We would also like to mention also the work of Kotani and Simon [16] and Schulz-Baldes [22] on random Schrödinger operators on the strip.…”

Section: Remarksmentioning

confidence: 70%

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: Remarksmentioning

confidence: 70%

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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“…We compute K s as a function of the endpoints of the support of the density h 0 , for the iid case when dµ n (λ) = h 0 (λ) dλ, for all n ∈ Z d . The computation of K s follows from [1], with an improvement in [27]. …”

Section: Exponential Localizationmentioning

confidence: 99%

Spectral and dynamical properties of random models with nonlocal and singular interactions

Hislop

Kirsch

Krishna

2005

Mathematische Nachrichten

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We give a spectral and dynamical description of certain models of random Schrödinger operators on L 2 (R d ) for which a modified version of the fractional moment method of Aizenman and Molchanov [3] can be applied. One family of models includes Schrödinger operators with random nonlocal interactions constructed from multidimensional wavelet bases. The second family includes Schrödinger operators with random singular interactions randomly located on sublattices of Z d , for d = 1, 2, 3. We prove that these models are amenable to Aizenman-Molchanov-type analysis of the Green's function, thereby eliminating the use of multiscale analysis. The basic technical result is an estimate on the expectation of fractional moments of the Green's function. Among our results, we prove a Wegner estimate, Hölder continuity of the integrated density of states, and spectral and Hilbert-Schmidt dynamical localization at negative energies.

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“…This has attracted some interest in the last decades as can be seen in the articles [15,16,19,20,21,23,24] dealing with discrete Schrödinger operators and [7] for the continuum case.…”

Section: Sparse Random Modelsmentioning

confidence: 99%

Absence of Continuous Spectral Types for Certain Non-Stationary Random Schrödinger Operators

Monvel

Stollmann

Stolz

2005

Ann. Henri Poincaré

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Abstract. We consider continuum random Schrödinger operators of the type Hω = −∆ + V0 + Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of −∆ + V0. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface ("random tube") in arbitrary dimension.

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Spectra of Anderson type models with decaying randomness

Cited by 6 publications

References 13 publications

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

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

Spectral and dynamical properties of random models with nonlocal and singular interactions

Absence of Continuous Spectral Types for Certain Non-Stationary Random Schrödinger Operators

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