Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials

TL;DR

This paper investigates the asymptotic behavior of the lowest eigenvalues of lattice-based random Schrödinger operators with unbounded potentials, demonstrating convergence to deterministic values and establishing a central limit theorem for fluctuations.

Contribution

It extends previous results to unbounded potentials, proving eigenvalue convergence and a multivariate CLT under new moment and regularity conditions.

Findings

Eigenvalues converge to deterministic limits as lattice spacing tends to zero.

Eigenvalue fluctuations follow a multivariate central limit theorem.

Results extend prior bounded potential analyses to unbounded cases.

Abstract

We consider random Schr\"odinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schr\"odinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials.