Spectral statistics for weakly correlated random potentials

TL;DR

This paper investigates the spectral properties of weakly correlated random Schrödinger operators, deriving estimates and analyzing spectral statistics in the localized phase for models with correlated potentials.

Contribution

It introduces new stochastic estimates and spectral analysis techniques for weakly correlated, alloy-type models with non-finite rank and sign-changing potentials.

Findings

Derived Wegner and Minami estimates for correlated potentials

Established spectral statistics in the localized phase

Analyzed models with long-range correlations

Abstract

We study localization and derive stochastic estimates (in particular, Wegner and Minami estimates) for the eigenvalues of weakly correlated random discrete Schr\"odinger operators in the localized phase. We apply these results to obtain spectral statistics for general discrete alloy type models where the single site perturbation is neither of finite rank nor of fixed sign. In particular, for the models under study, the random potential exhibits correlations at any range.