New characterizations of the region of complete localization for random Schrödinger operators

Abstract

We study the region of complete localization in a class of random operators which includes random Schrödinger operators with Anderson-type potentials and classical wave operators in random media, as well as the Anderson tight-binding model. We establish new characterizations or criteria for this region of complete localization, given either by the decay of eigenfunction correlations or by the decay of Fermi projections. (These are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region.) Using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity.

References in corpus (4)

• Moment Analysis for Localization in Random Schroedinger Operators
• Equality of the bulk and edge Hall conductances in a mobility gap
• Linear response theory for magnetic Schroedinger operators in disordered media
• Simplicity of eigenvalues in the Anderson model

Cited by in corpus (40)

• Localization Bounds for Multiparticle Systems
• A comprehensive proof of localization for continuous Anderson models with singular random potentials
• Linear response theory for magnetic Schroedinger operators in disordered media
• Simplicity of eigenvalues in the Anderson model
• Generalized eigenvalue-counting estimates for the Anderson model
• Perturbation theory for the Nonlinear Schroedinger Equation with a random potential
• The bootstrap multiscale analysis for the multi-particle Anderson model
• Localisation for non-monotone Schroedinger operators
• Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas
• Quantization of the Hall conductance and delocalization in ergodic Landau Hamiltonians
• An eigensystem approach to Anderson localization
• Quantization of edge currents along magnetic barriers and magnetic guides
• Characterization of the Anderson metal-insulator transition for non ergodic operators and application
• Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models
• Decorrelation estimates for the eigenlevels of the discrete Anderson model in the localized regime
• The conductivity measure for the Anderson model
• Spectral statistics for the discrete Anderson model in the localized regime
• Persistence of Anderson localization in Schrödinger operators with decaying random potentials
• Localization for random block operators
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Conductivity and the current-current correlation measure
• Bootstrap multiscale analysis and localization for multi-particle continuous Anderson Hamiltonians
• Characterization of the metal-insulator transport transition for the two-particle Anderson model
• Multiplicity bound of Singular Spectrum for higher rank Anderson models
• Ac-conductivity and electromagnetic energy absorption for the Anderson model in linear response theory
• Wegner estimate and disorder dependence for alloy-type Hamiltonians with bounded magnetic potential
• Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model
• Eigensystem multiscale analysis for the Anderson model via the Wegner estimate
• Unique continuation principle for spectral projections of Schr\" odinger operators and optimal Wegner estimates for non-ergodic random Schr\" odinger operators
• Many-body localization in the droplet spectrum of the random XXZ quantum spin chain
• A bound on the averaged spectral shift function and a lower bound on the density of states for random Schrödinger operators on ${\mathbb{R}}^d$
• Random Schrödinger Operators on discrete structures
• An eigensystem approach to Anderson localization for multi-particle systems
• Wegner estimate for discrete Schrödinger operators with Gaussian random potentials
• Splitting of the Landau levels by magnetic perturbations and Anderson transition in 2D-random magnetic media
• Eigensystem multiscale analysis for Anderson localization in energy intervals
• The Liouville equation for singular ergodic magnetic Schrödinger operators
• Dynamical mobility edge for various random Landau Hamiltonians
• Global multiplicity bounds and Spectral Statistics Random Operators
• Density of states and Delocalization for discrete magnetic random Schrödinger operators