Characterization of the Anderson metal-insulator transition for non ergodic operators and application

TL;DR

This paper extends the analysis of the Anderson metal-insulator transition to non-ergodic operators, particularly Delone-Anderson potentials, demonstrating localization properties and the existence of a mobility edge in a more general setting.

Contribution

It reformulates the Bootstrap Multiscale Analysis for non-ergodic operators and establishes localization results for Delone-Anderson potentials, advancing understanding beyond ergodic cases.

Findings

Proved localization for non-ergodic Delone-Anderson potentials.

Established uniform Wegner estimates in the non-ergodic setting.

Demonstrated the existence of a mobility edge in the Landau operator with Delone-Anderson potential.

Abstract

We study the Anderson metal-insulator transition for non ergodic random Schr\"odinger operators in both annealed and quenched regimes, based on a dynamical approach of localization, improving known results for ergodic operators into this more general setting. In the procedure, we reformulate the Bootstrap Multiscale Analysis of Germinet and Klein to fit the non ergodic setting. We obtain uniform Wegner Estimates needed to perform this adapted Multiscale Analysis in the case of Delone-Anderson type potentials, that is, Anderson potentials modeling aperiodic solids, where the impurities lie on a Delone set rather than a lattice, yielding a break of ergodicity. As an application we study the Landau operator with a Delone-Anderson potential and show the existence of a mobility edge between regions of dynamical localization and dynamical delocalization.