From fixed-energy MSA to dynamical localization: A continuing quest for elementary proofs

TL;DR

This paper reviews techniques for proving Anderson localization, presenting a short, elementary derivation of dynamical localization from fixed-energy Green function analysis, applicable to multi-particle models.

Contribution

It offers a simplified, elementary proof of dynamical localization using fixed-energy analysis, extending previous methods to multi-particle systems.

Findings

Elementary derivation of dynamical localization

Quantitative estimates for Green functions

Applicability to multi-particle models

Abstract

We review several techniques and ideas initiated by a remarkable work by Spencer [26], used and further developed in numerous subsequent researches. We also describe a relatively short and elementary derivation of the spectral and strong dynamical Anderson localization from the fixed-energy analysis of the Green functions, obtained either by the Multi-Scale Analysis (MSA) or by the Fractional-Moment Method (FMM). This derivation goes in the same direction as the Simon--Wolf criterion [28], but provides quantitative estimates, applies also to multi-particle models and, combined with a simplified variant of the Germinet--Klein argument [20], results in an elementary proof of dynamical localization.