Sharp localization on the first supercritical stratum for Liouville frequencies

TL;DR

This paper proves Anderson localization for Schrödinger operators with analytic potentials at Liouville frequencies in a specific supercritical energy regime, extending previous methods and introducing new techniques for sharp analysis.

Contribution

It establishes localization results for Liouville frequencies using enhanced large deviation estimates and complexity bounds, advancing the understanding of spectral properties in this regime.

Findings

Proves Anderson localization for Liouville frequencies in the supercritical regime.

Extends large deviation estimates to weak Liouville frequencies.

Introduces new methods for sharp analysis of Liouville frequencies.

Abstract

We establish Anderson localization for Schr\"odinger operators with even analytic potentials on the first supercritical stratum for Liouville frequencies in the sharp regime $\{E: L(\omega,E)>\beta(\omega)>0, \kappa(\omega,E)=1\}$, with $\kappa(\omega,E)$ being Avila's acceleration. This paper builds on the large deviation measure estimate and complexity bound scheme, originally developed for Diophantine frequencies by Bourgain, Goldstein and Schlag \cites{BG,BGS1,BGS2}, and the improved complexity bounds in \cite{HS1}. Additionally, it strengthens the large deviation estimates for weak Liouville frequencies in \cite{HZ}. We also introduce new ideas to handle Liouville frequencies in a sharp way.