Anderson Localization at Band Edges for Random Magnetic Fields

TL;DR

This paper demonstrates Anderson localization at the edges of broadened Landau bands for a 2D magnetic Schrödinger operator with combined deterministic and random magnetic fields, using multiscale analysis and Wegner estimates.

Contribution

It extends localization results to a model with both deterministic periodic and random magnetic fields, including band edge analysis.

Findings

Spectrum consists of broadened Landau bands.

Band edges exhibit pure point spectrum with exponential localization.

Localization proven using multiscale analysis and Wegner estimates.

Abstract

We consider a magnetic Schr\"odinger operator in two dimensions. The magnetic field is given as the sum of a large and constant magnetic field and a random magnetic field. Moreover, we allow for an additional deterministic potential as well as a magnetic field which are both periodic. We show that the spectrum of this operator is contained in broadened bands around the Landau levels and that the edges of these bands consist of pure point spectrum with exponentially decaying eigenfunctions. The proof is based on a recent Wegner estimate obtained in \cite{EH2} and a multiscale analysis.