An eigensystem approach to Anderson localization

TL;DR

This paper presents a novel eigensystem-based multiscale analysis method to prove Anderson localization at high disorder, avoiding traditional Green's function techniques and enabling simultaneous treatment of all energies.

Contribution

It introduces a new eigensystem multiscale analysis approach that simplifies proving localization for the Anderson model at high disorder.

Findings

Establishes level spacing and localization with high probability

Treats all energies simultaneously in finite volume analysis

Labels eigenvalues and eigenfunctions by site locations

Abstract

We introduce a new approach for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). Information about eigensystems at a given scale is used to derive information about eigensystems at larger scales. This eigensystem multiscale analysis treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions in a fixed box with high probability. A new feature is the labeling of the eigenvalues and eigenfunctions by the sites of the box.