The integrated density of states for some random operators with nonsign definite potentials
type: Item
instance of: scholarly article
title: The integrated density of states for some random operators with nonsign definite potentials
zbMATH Open document ID: 1013.60046
author: Peter D. Hislop
author: Frédéric Klopp
published in: Journal of Functional Analysis
publication date: 2003-03-26T00:00:00Z
review text: The authors investigate the integrated density of states (IDS) for random Anderson-type additive and multiplicative perturbations of deterministic background operators with nonsign definite potentials. One of the features is that for such operators the single-site potential does not have a fixed sign. In fact, the unperturbed background medium in the multiplicative case is described by a divergence form operator \[ H_0^M = - C_0 \rho_0^{1/2} \nabla \cdot \rho_0^{-1} \nabla \rho_0^{-1/2} C_0, \] where \(\rho_0\) and \(C_0\) are positive functions that describe the unperturbed density and sound velocity, respectively. They are assumed to be sufficiently regular so that \(C_0^{\infty} (\mathbb R^d)\) is an operator core for \(H_0^M\). While, the unperturbed background medium in the additive case is described by a Schrödinger operator \(H_0\) given by \(H_0^A = ( - i \nabla - A)^2 + W\), where \(A\) is a vector potential with \(W = W_+ - W_-\) and \(W_+\) (resp. \(W_-\)) is a member of the local Kato (resp. Kato) class. The local perturbation in the Anderson-type Alloy model is defined by the following random potentials \[ V_{\Lambda}(x) = \sum_{ i \in \widetilde{\Lambda}} \lambda_i(\omega) u_i ( x - i - \xi_i(\omega')), \] where \(\widetilde{\Lambda}\) denotes the lattice points in the region \(\Lambda\), and \(\xi_i\)'s are uniformly bounded random variables modeling thermal vibrations, and \(\lambda_i\)'s are random variables compactly supported in a neighborhood of the origin. The following hypotheses are assumed: (H1) The selfadjoint operator \(H_0^X\) (for \(X=A\) or \(M\)) is essentially selfadjoint on \(C_0^{\infty}(\mathbb R^d)\) and is semi-bounded and has an open spectral gap. (H2) \(H_0^X\) is locally compact for \(X=A\) or \(M\). (H3) The single-site potential \(u_k\) is continuous and compactly supported, and \(\sum_k |u_k |< C_V\) for any bounded \(\Lambda \subset \mathbb R^d\), and the family \(\{ \|u_k \|_{\infty} \}_{k \in Z^d}\) is also uniformly boun…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 47B80
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 47N55
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 1888340
zbMATH Keywords: integrated density of states
zbMATH Keywords: Anderson-type random potentials
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: Wegner estimate
zbMATH Keywords: localization
zbMATH Keywords: monotonic variation
MaRDI profile type: Publication
full work available at URL: jfan.2002.3947
OpenAlex ID: W2043560168
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DOI: 10.1006/JFAN.2002.3947
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description: scientific article; zbMATH DE number 1888340