New characterizations of the region of complete localization for random Schr\"odinger operators

TL;DR

This paper introduces new criteria based on eigenfunction correlations and Fermi projection decay to characterize the complete localization region in various random Schrödinger operators, providing necessary and sufficient conditions.

Contribution

It offers novel characterizations of the complete localization region, linking decay properties to spectral and eigenfunction features in random operators.

Findings

Eigenvalues have finite multiplicity in the localization region.

Decay of eigenfunction correlations characterizes localization.

Fermi projection decay is a necessary and sufficient condition.

Abstract

We study the region of complete localization in a class of random operators which includes random Schr\"odinger 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.