Eigensystem multiscale analysis for the Anderson model via the Wegner estimate

TL;DR

This paper introduces a new eigensystem multiscale analysis method for the Anderson model that leverages the Wegner estimate, enabling simultaneous localization proof across energy intervals without restrictive assumptions.

Contribution

The paper develops a novel EMSA approach based on the Wegner estimate, removing previous level spacing restrictions and accommodating H"older continuous distributions.

Findings

Proves localization for the Anderson model with H"older continuous distributions.

Establishes pure point spectrum with exponentially decaying eigenfunctions.

Demonstrates dynamical localization in the model.

Abstract

We present a new approach to the eigensystem multiscale analysis (EMSA) for random Schr\"odinger operators that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the energy interval with high probability. It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization). The new method removes the restrictive level spacing hypothesis used in the previous versions of the EMSA. The method is presented in the context of the Anderson model, allowing for single site probability distributions that are H\"older continuous of order $\alpha \in (0,1]$.