Band Edge Localization Beyond Regular Floquet Eigenvalues

Albrecht Seelmann,Matthias Täufer

doi:10.1007/s00023-020-00911-7

Abstract

We prove that localization near band edges of multi-dimensional ergodic random Schrödinger 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.

Highlights

The Anderson model dates back to the work of Anderson in 1958 [2] in condensed matter physics who argued that the presence of disorder will drastically change the dynamics of electrons in a solid
We refer to the monographs [3,38,42,48] for an overview on the mathematics literature
The prototypical model investigated in this context is the ergodic Alloy-type or continuum Anderson model

Summary

Introduction and Results

The Anderson model dates back to the work of Anderson in 1958 [2] in condensed matter physics who argued that the presence of disorder will drastically change the dynamics of electrons in a solid. Klopp proved that Lifshitz tails occur for the random operator Hωerg if the background operator Hper has regular Floquet eigenvalues near these edges, which means that these edges are generated by a quadratic extremum of an eigenvalue curve in the so-called dispersion relation This implies an initial scale estimate which, together with a Wegner estimate, can be used to start the multi-scale analysis and prove localization [47]. Therein it is shown in two space dimensions that even if a proper Lifshitz tail does not occur for the integrated density of states of the random operator, a weaker version of (1.4) with −d/2 replaced by −α for some α > 0 always holds Such an asymptotic still implies an initial scale estimate and localization, see Theorems 0.3 and 0.4 in [33].

The Model

Main Results

An Initial Scale Estimate for Non-ergodic Random Schrodinger Operators