On Eigenvalue spacings for the 1-D Anderson model with singular site distribution

TL;DR

This paper investigates eigenvalue spacings and local eigenvalue statistics for 1D lattice Schrödinger operators with singular potentials, establishing Poisson statistics and Minami's inequality using Furstenberg measure properties.

Contribution

It introduces new regularity results for Furstenberg measures and density of states, extending eigenvalue spacing analysis to singular potential distributions.

Findings

Established Poisson statistics for eigenvalue spacings.

Proved a version of Minami's inequality for singular potentials.

Connected eigenvalue statistics with Furstenberg measure properties.

Abstract

We study eigenvalue spacings and local eigenvalue statistics for 1D lattice Schrodinger operators with Holder regular potential, obtaining a version of Minami's inequality and Poisson statistics for the local eigenvalue spacings. The main additional new input are regular properties of the Furstenberg measures and the density of states obtained in some of the author's earlier work.