Eigensystem multiscale analysis for Anderson localization in energy intervals

TL;DR

This paper introduces a novel eigensystem multiscale analysis method to prove localization in the Anderson model within specific energy intervals, focusing on eigenfunctions and eigenvalues without relying on Green's functions.

Contribution

It develops a new multiscale analysis approach that treats all energies simultaneously and establishes localization in energy intervals at the bottom of the spectrum.

Findings

Proves localization for the Anderson model in an energy interval.

Establishes level spacing and exponential decay of eigenfunctions.

Provides a new framework avoiding Green's function analysis.

Abstract

We present an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability. 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). In any given scale we only have decay for eigenfunctions with eigenvalues in the energy interval, and no information about the other eigenfunctions. For this reason, going to a larger scale requires new arguments that were not necessary in our previous eigensystem multiscale analysis for the Anderson model at high disorder, where in a given scale we have decay for all eigenfunctions.