Simplicity of eigenvalues in Anderson-type models

TL;DR

This paper proves that eigenvalues are almost surely simple in various Anderson-type random Schrödinger operators, extending Simon’s methods beyond the localization regime and providing general criteria for eigenvalue simplicity.

Contribution

It introduces new criteria for eigenvalue simplicity applicable to a broad class of Anderson models, including matrix-valued and finite-support potentials, beyond previous localization-focused results.

Findings

Eigenvalues are almost surely simple across the spectrum.

Methods extend to models with matrix-valued potentials.

Criteria exclude local and global symmetries, ensuring simplicity.

Abstract

We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schr\"odinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.