Global multiplicity bounds and Spectral Statistics Random Operators

TL;DR

This paper establishes lower bounds on spectral multiplicity for Anderson operators with full support randomness and links higher multiplicity to non-Poisson local spectral statistics, advancing understanding of spectral behavior in disordered systems.

Contribution

It demonstrates a uniform lower bound on spectral multiplicity for certain Anderson models and connects multiplicity with local spectral statistics, especially non-Poisson behavior.

Findings

Spectral multiplicity has a uniform lower bound on sets of positive measure.

Higher spectral multiplicity implies non-Poisson local spectral statistics.

In higher rank Anderson models, multiplicity greater than one correlates with non-Poisson statistics.

Abstract

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on $\mathbb{R}$. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory. In particular, in higher rank Anderson models with pure-point spectrum, with the randomness having support equal to $\mathbb{R}$, there is a uniform lower bound on spectral multiplicity and in case this is larger than one the local statistics is not Poisson.