Localization for random perturbations of periodic Schroedinger operators with regular Floquet eigenvalues

TL;DR

This paper proves a localization result for a class of ergodic Schrödinger operators with periodic background and Anderson-type perturbations, under conditions on Floquet eigenvalues at spectral edges.

Contribution

It establishes localization at spectral edges for operators with regular Floquet eigenvalues, extending understanding of spectral properties under perturbations.

Findings

Pure point spectrum in an interval near the spectral edge for almost all realizations.

Localization holds under the condition of positive definite Hessian of Floquet eigenvalues.

Results apply to nonnegative Anderson-type perturbations of periodic Schrödinger operators.

Abstract

We prove a localization theorem for continuous ergodic Schr\"odinger operators $ H_\omega := H_0 + V_\omega $, where the random potential $ V_\omega $ is a nonnegative Anderson-type perturbation of the periodic operator $ H_0$. We consider a lower spectral band edge of $ \sigma (H_0) $, say $ E= 0 $, at a gap which is preserved by the perturbation $ V_\omega $. Assuming that all Floquet eigenvalues of $ H_0$, which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval $ I $ containing 0 such that $ H_\omega $ has only pure point spectrum in $ I $ for almost all $ \omega $.