Eigensystem Bootstrap Multiscale Analysis for the Anderson Model

TL;DR

This paper introduces an improved multiscale analysis technique for the Anderson model that enhances localization proofs by focusing on eigensystems rather than Green's functions, achieving exponential localization with high probability.

Contribution

It develops a bootstrap eigensystem multiscale analysis that simplifies and strengthens localization proofs for the Anderson model at high disorder levels.

Findings

Proves pure point spectrum with exponentially decaying eigenfunctions.

Establishes exponential localization in finite volume boxes.

Achieves high probability bounds for eigenfunction localization.

Abstract

We use a bootstrap argument to enhance the eigensystem multiscale analysis, introduced by Elgart and Klein for proving localization for the Anderson model at high disorder. The eigensystem multiscale analysis studies finite volume eigensystems, not finite volume Green's functions. It yields pure point spectrum with exponentially decaying eigenfunctions and dynamical localization. The starting hypothesis for the eigensystem bootstrap multiscale analysis only requires the verification of polynomial decay of the finite volume eigenfunctions, at some sufficiently large scale, with some minimal probability independent of the scale. It yields exponential localization of finite volume eigenfunctions in boxes of side $L$, with the eigenvalues and eigenfunctions labeled by the sites of the box, with probability higher than $1-\mathrm{e}^{-L^\xi}$, for any desired $0<\xi<1$.