Band edge localization beyond regular Floquet eigenvalues

TL;DR

This paper establishes universal localization near band edges for multi-dimensional ergodic random Schrödinger operators with periodic backgrounds, using novel initial scale estimates that do not rely on Floquet theory.

Contribution

It introduces a new approach to prove band edge localization that bypasses Floquet theory and applies to non-ergodic settings, broadening the understanding of localization phenomena.

Findings

Localization at band edges is universal under broad conditions.

The proof employs unique continuation and large deviation estimates instead of Floquet theory.

The approach extends to non-ergodic cases, indicating wider applicability.

Abstract

We prove that localization near band edges of multi-dimensional ergodic random Schr\"odinger operators with periodic background potential in $L^2(\mathbb{R}^d)$ is universal. By this we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.