Sharp localization on the first supercritical stratum for Liouville frequencies

2024/05/13 by Rui Han, Han, Rui · 1 citation
Earth and Planetary Sciences · Environmental Science · #Seismic Imaging and Inversion Techniques #Methane Hydrates and Related Phenomena #High-pressure geophysics and materials

paper · pdf · doi:10.48550/arxiv.2405.07810

Abstract

We establish Anderson localization for Schrödinger operators with even analytic potentials on the first supercritical stratum for Liouville frequencies in the sharp regime \E: L(ω,E)>β(ω)>0, κ(ω,E)=1\, with κ(ω,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 \citesBG,BGS1,BGS2, and the improved complexity bounds in \citeHS1. Additionally, it strengthens the large deviation estimates for weak Liouville frequencies in \citeHZ. We also introduce new ideas to handle Liouville frequencies in a sharp way.

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• Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices

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