Generalized Eigenvalue-Counting Estimates for the Anderson Model

Combes, Jean‐Michel; Germinet, François; Klein, Abel

doi:10.1007/s10955-009-9731-3

Cited by 58 publications

(69 citation statements)

References 30 publications

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“…Extensions to k > 2 were subsequently presented by Bellissard, Hislop, and Stolz [2], Graf and Vaghi [11], and Combes, Germinet, and Klein [6]. In particular, our derivation of Theorem 2 has benefitted from the strategy of [6].…”

Section: (Mean Density Of States) For Any Intervalmentioning

confidence: 90%

Matrix regularizing effects of Gaussian perturbations

Aizenman

Peled

Schenker

et al. 2017

Commun. Contemp. Math.

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The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for H = A + V , where A is the base matrix and V is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of H in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of H −1 and for the distribution of the norm of H −1 applied to a fixed vector. The bounds are uniform in A and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.

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Section: (Mean Density Of States) For Any Intervalmentioning

confidence: 90%

Matrix regularizing effects of Gaussian perturbations

Aizenman

Peled

Schenker

et al. 2017

Commun. Contemp. Math.

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“…In particular, the general m-level Wegner estimate is known to hold for essentially all distributions µ of the random potential v(x); see [2,11]. We refer the reader to [5] for the state of the art results concerning eigenvalue counting inequalities for H A . However, the current understanding of many (in fact almost all) other random models of interest remains partial at best.…”

Section: Previous Resultsmentioning

confidence: 99%

“…In this model, the evolution of the wave function ψ on the d-dimensional lattice Z d is given by the Schrödinger equation 5) where the self-adjoint Hamiltonian H is a sum of the hopping term H 0 and the potential V , of the form…”

Section: Application To Random Schrödinger Operatorsmentioning

confidence: 99%

Eigenvalue counting inequalities, with applications to Schrödinger operators

Elgart¹,

Schmidt²

2015

J. Spectr. Theory

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ABSTRACT. We derive a sufficient condition for a Hermitian N × N matrix A to have at least m eigenvalues (counting multiplicities) in the interval (−ǫ, ǫ). This condition is expressed in terms of the existence of a principal (N − 2m) × (N − 2m) submatrix of A whose Schur complement in A has at least m eigenvalues in the interval (−Kǫ, Kǫ), with an explicit constant K.We apply this result to a random Schrödinger operator H ω , obtaining a criterion that allows us to control the probability of having m closely lying eigenvalues for H ω -a result known as an m-level Wegner estimate. We demonstrate its usefulness by verifying the input condition of our criterion for some physical models. These include the Anderson model and random block operators that arise in the Bogoliubov-de Gennes theory of dirty superconductors.

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“…Therefore, we will use the results of [13], [22] and [21] (see also [14]). We first recall the Minami estimate satisfied by H ω,L (see, e.g., [8] and references therein): there exists C > 0 such that, for I ⊂ R, one has…”

Section: Resonances In the Random Casementioning

confidence: 99%

Resonances for large one-dimensional “ergodic” systems

Klopp

2016

Anal. PDE

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Abstract. The present paper is devoted to the study of resonances for one-dimensional quantum systems with a potential that is the restriction to some large box of an ergodic potential. For discrete models both on a half-line and on the whole line, we study the distributions of the resonances in the limit when the size of the box where the potential does not vanish goes to infinity. For periodic and random potentials, we analyze how the spectral theory of the limit operator influences the distribution of the resonances.Résumé. Dans cet article, nousétudions les résonances d'un système unidimensionnel plongé dans un potentiel qui est la restrictionà un grand intervalle d'un potentiel ergodique. Pour des modèles discrets sur la droite et la demie droite, nousétudions la distribution des résonances dans la limite de la taille de boîte infinie. Pour des potentiels périodiques et aléatoires, nous analysons l'influence de la théorie spectrale de l'opérateur limite sur la distribution des résonances. IntroductionConsider V : Z → R a bounded potential and, on ℓ 2 (Z), the Schrödinger operator H = −∆ + V defined by (Hu)(n) = u(n + 1) + u(n − 1) + V (n)u(n), ∀n ∈ Z, for u ∈ ℓ 2 (Z). The potentials V we will deal with are of two types:• V periodic;• V = V ω , the random Anderson model, i.e., the entries of the diagonal matrix V are independent identically distributed non constant random variable. The spectral theory of such models has been studied extensively (see, e.g., [19]) and it is well known that• when V is periodic, the spectrum of H is purely absolutely continuous;• when V = V ω is random, the spectrum of H is almost surely pure point, i.e., the operator only has eigenvalues; moreover, the eigenfunctions decay exponentially at infinity. Pick L ∈ N * . The main object of our study is the operator (0.1)For L large, the operator H L is a simple Hamiltonian modeling a large sample of periodic or random material in the void. It is well known in this case (see, e.g., [43]) that not only does the spectrum 2010 Mathematics Subject Classification. 47B80, 47H40, 60H25, 82B44, 35B34.

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Generalized Eigenvalue-Counting Estimates for the Anderson Model

Cited by 58 publications

References 30 publications

Matrix regularizing effects of Gaussian perturbations

Matrix regularizing effects of Gaussian perturbations

Eigenvalue counting inequalities, with applications to Schrödinger operators

Resonances for large one-dimensional “ergodic” systems

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