Localization for the random displacement model
type: Item
instance of: scholarly article
title: Localization for the random displacement model
zbMATH Open document ID: 1285.82030
DOI: 10.1215/00127094-1548353
author: Frédéric Klopp
author: Michael Loss
author: S. Nakamura
author: Günter Stolz
published in: Duke Mathematical Journal
publication date: 2012-04-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.2483
full work available at URL: https://projecteuclid.org/euclid.dmj/1330610808
review text: The random displacement model is a random Schrödinger operator in the continuum with a random perturbation of the periodic potential. In this article, the authors prove spectral localization (a.s. pure point spectrum and exponentially decaying eigenfunctions) and dynamical localization in an interval above the ground state energy. A key ingredient is previous work by two of the authors characterizing the configurations of random displacements that minimize the ground state energy. This is turn is derived from a careful analysis of the ground state energy of Schrödinger operators in a box with von Neumann boundary conditions, which shows that the potential that minimizes the ground state energy is centered in a corner. The structure of the localization proof is standard: (i) smallness of the integrated density of states and (ii) Wegner estimate, used as inputs to the multiscale method. While (i) \& (ii) are proved in details in the paper and both use the a priori knowledge on the ground state energy, the running of the multiscale method from them is known and only briefly discussed for pedagogical reasons.
reviewed by: S. Bachmann
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 47B80
zbMATH DE Number: 6024992
zbMATH Keywords: Lifshitz tail
zbMATH Keywords: Wegner estimate
zbMATH Keywords: localization
MaRDI profile type: Publication
arXiv ID: 1007.2483
cites work: Minimizing the ground state energy of an electron in a randomly deformed lattice
cites work: Low energy properties of the random displacement model
cites work: Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators
cites work: The \(L^p\)-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators.
cites work: Lower transport bounds for one-dimensional continuum Schrödinger operators
cites work: Localization for one-dimensional, continuum, Bernoulli-Anderson models.
cites work: Power-law bounds on transfer matrices and quantum dynamics in one dimension
cites work: Brownian survival and Lifshitz tail in perturbed lattice disorder
cites work: Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice
cites work: Q5436574
cites work: Localization for Schrödinger operators with Poisson random potential
cites work: Bootstrap multiscale analysis and localization in random media
cites work: Localization for the random displacement model at weak disorder
cites work: The integrated density of states for some random operators with nonsign definite potentials
cites work: Delocalization in random polymer models
cites work: Q4110164
cites work: Multiscale Analysis and Localization of Random Operators
cites work: Q4836378
cites work: Spectral extrema and Lifshitz tails for non-monotonous alloy type models
cites work: Lifshitz tails for generalized alloy-type random Schrödinger operators
cites work: Anderson localization for 2D discrete Schrödinger operators with random magnetic fields
cites work: Spectral averaging and the Krein spectral shift
cites work: Caught by disorder. Bound states in random media
cites work: Q3996935
description: scientific article; zbMATH DE number 6024992