Correlations Estimates in the Lattice Anderson Model

TL;DR

This paper introduces a new proof technique for correlation estimates in the lattice Anderson model, extending previous results and deriving new bounds on eigenvalue correlations and level statistics in disordered quantum systems.

Contribution

It provides a generalized correlation estimate for moments of the resolvent, leading to new Wegner-type bounds and results on eigenvalue statistics in the strong localization regime.

Findings

Extended correlation estimates for resolvent moments.

Derived new Wegner-type eigenvalue bounds.

Confirmed Poisson statistics and eigenvalue simplicity in localization regime.

Abstract

We give a new proof of correlation estimates for arbitrary moments of the resolvent of random Schr\"odinger operators on the lattice that generalizes and extends the correlation estimate of Minami for the second moment. We apply this moment bound to obtain a new $n$-level Wegner-type estimate that measures eigenvalue correlations through an upper bound on the probability that a local Hamiltonian has at least $n$ eigenvalues in a given energy interval. Another consequence of the correlation estimates is that the results on the Poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation-invariant, selfadjoint, lattice operators with decaying off-diagonal terms and random potentials.