Eigenvalue statistics for random Schrodinger operators with non rank one perturbations

TL;DR

This paper demonstrates that local eigenvalue statistics for generalized lattice and continuum Anderson models with finite-rank perturbations follow a compound Poisson distribution, with the Levy measure supported on a finite set determined by the perturbation rank.

Contribution

It establishes the distribution of local eigenvalue statistics for finite-rank perturbed Anderson models, extending previous results to non-rank-one perturbations and continuum models.

Findings

Eigenvalue statistics are compound Poisson distributed.

Levy measure support is finite and determined by the perturbation rank.

Results apply to both lattice and continuum Anderson models.

Abstract

We prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice Anderson models constructed with finite-rank perturbations are compound Poisson distributed. This distribution is characterized by the fact that the Levy measure is supported on at most a finite set determined by the rank. The proof relies on a Minami-type estimate for finite-rank perturbations. For Anderson-type continuum models on $\R^d$, we prove a similar result for certain natural random variables associated with the local eigenvalue statistics. We prove that the compound Poisson distribution associated with these random variables has a Levy measure whose support is at most the set of positive integers.