Limiting distribution of extremal eigenvalues of d-dimensional random Schr\"odinger operator

TL;DR

This paper studies the extremal eigenvalues of a d-dimensional random Schrödinger operator with decaying potential, showing their distribution converges to a Poisson process under certain conditions, with different behaviors depending on potential decay.

Contribution

It establishes the limiting distribution of extremal eigenvalues for random Schrödinger operators with decaying potentials, including conditions for Poisson statistics and border-line cases.

Findings

IDS matches free Laplacian in general cases

Extremal eigenvalues converge to an inhomogeneous Poisson process

Decay of potential affects eigenvalue statistics

Abstract

We consider Schr\"odinger operator with random decaying potential on $\ell^2 ({\bf Z}^d)$ and showed that, (i) IDS coincides with that of free Laplacian in general cases, and (ii) the set of extremal eigenvalues, after rescaling, converges to a inhomogeneous Poisson process, under certain condition on the single-site distribution, and (iii) there are "border-line" cases, such that we have Poisson statistics in the sense of (ii) above if the potential does not decay, while we do not if the potential does decay.