Generalized eigenvalue-counting estimates for the Anderson model

TL;DR

This paper extends eigenvalue-counting estimates for the Anderson model to multiple eigenvalues and arbitrary intervals, providing new insights into eigenvalue multiplicity and ac-conductivity under broad conditions.

Contribution

It generalizes Minami's estimate to n eigenvalues over arbitrary intervals with atomless probability measures, advancing understanding of spectral properties.

Findings

New eigenvalue multiplicity bounds

Validation of Mott's formula for H"older continuous distributions

Extended eigenvalue estimates to multiple intervals

Abstract

We generalize Minami's estimate for the Anderson model and its extensions to $n$ eigenvalues, allowing for $n$ arbitrary intervals and arbitrary single-site probability measures with no atoms. As an application, we derive new results about the multiplicity of eigenvalues and Mott's formula for the ac-conductivity when the single site probability distribution is H\"older continuous.