Lifshitz asymptotics and localization for random breather models

TL;DR

This paper establishes Lifshitz asymptotics and localization phenomena at low energies for non-negative random potentials, including breather models, using multi-scale analysis and Wegner estimates, even in non-ergodic settings.

Contribution

It proves Lifshitz behavior for breather potentials and extends the results to all non-negative random potentials, providing a comprehensive analysis of localization without requiring ergodicity.

Findings

Lifshitz asymptotics at the spectrum's bottom

Localization via multi-scale analysis

Extension to non-ergodic models

Abstract

We prove Lifshitz behavior at the bottom of the spectrum for non--negative random potentials, i.\,e.\ show that the IDS is exponentially small at low energies. The theory is developed for the breather potential and generalized to all non--negative random potentials in a second step. Since our models need not be ergodic, we need to identify the minimum of the spectrum. We deduce an initial length scale estimate from the Lifshitz bound for the breather model and combine it with a recent Wegner estimate to establishes Anderson localization via multi-scale analysis. Finally, for ergodic models, we complement the Lifshitz behavior with a lower bound. We provide detailed proofs accessible to non-experts.