Eigenfunction Statistics in the Localized Anderson Model

Killip, Rowan; Nakano, Fumihiko

doi:10.1007/s00023-006-0298-0

Cited by 39 publications

(48 citation statements)

References 11 publications

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“…2). Theorem 0.3 is the analogue for resonances of the well known result on the distribution of eigenvalues and localization centers for the Anderson model in the localized phase (see, e.g., [31,17,13]). As in the case of the periodic potential, Theorem 0.3 only describes the resonances closest to the real axis.…”

Section: Introductionmentioning

confidence: 84%

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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Section: Introductionmentioning

confidence: 84%

Resonances for large one-dimensional “ergodic” systems

Klopp

2016

Anal. PDE

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“…The main value of [Min96] is the introduction of a flexible estimate that establishes the existence of a gap between two subsequent eigenvalues, an estimate that is now called the Minami estimate. The first result on the convergence of point processes of both the eigenvalues and the concentration centres of the eigenfunctions is [KilNak07]; see also [Nak07]. The currently strongest available results are in [GerKlo14] and [GerKlo13], where [GerKlo14] works in the bulk of the spectrum and [GerKlo13] close to the top; see also [GerKlo11].…”

Section: Relation To Anderson Localisationmentioning

confidence: 99%

The Parabolic Anderson Model

König

2016

Pathways in Mathematics

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Each "Pathways in Mathematics" book offers a roadmap to a currently well developing mathematical research field and is a first-hand information and inspiration for further study, aimed both at students and researchers. It is written in an educational style, i.e., in a way that is accessible for advanced undergraduate and graduate students. It also serves as an introduction to and survey of the field for researchers who want to be quickly informed about the state of the art. The point of departure is typically a bachelor/masters level background, from which the reader is expeditiously guided to the frontiers. This is achieved by focusing on ideas and concepts underlying the development of the subject while keeping technicalities to a minimum. Each volume contains an extensive annotated bibliography as well as a discussion of open problems and future research directions as recommendations for starting new projects More information about this series at

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“…We have, ≤ · · · ≤ e ω,k |B k | , are the eigenvalues of H ω k l 2 (B k ), μ ω k is the corresponding normalized counting measure given by (1.1) and ξ ω,e k is the rescaled measure near e given by (1.2). We refer the reader to [18] for a discussion of the regime where both space and energy are rescaled. The averaged spectral measure for H ω is given by (3.3) and the Wegner estimate yields that μ av has a bounded density η(t) with respect to L. A basic result for the Anderson model is that for P-a.e.…”

Section: Vol 9 (2008) Poisson Statistics Of Eigenvalues 701mentioning

confidence: 99%

“…He proved Poisson statistics of eigenvalues in the localized regime [18,25]. Minami's method has its origins in Molchanov's paper [26], where the first rigorous proof of the absence of energy level repulsion is given for a continuous one-dimensional model.…”

Section: Introductionmentioning

confidence: 99%

“…Minami's method has its origins in Molchanov's paper [26], where the first rigorous proof of the absence of energy level repulsion is given for a continuous one-dimensional model. After Minami's paper [25], the technique and its variations have been used to prove Poisson statistics of eigenvalues for different models [3,4,18,19,30]. In this paper, we combine existing and new results to prove Poisson statistics of eigenvalues for the hierarchical Anderson model (the precise definition of the model and the statement of our results are given in Section 3).…”

Section: Introductionmentioning

confidence: 99%

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