Anderson localization and H\"older continuity of the integrated density of states for analytic quasiperiodic Schr\"odinger operators

TL;DR

This paper proves Anderson localization and H"older continuity of the integrated density of states for analytic quasiperiodic Schr"odinger operators with Diophantine frequencies, introducing a novel multi-scale analysis method.

Contribution

It develops a new multi-scale analysis technique applicable to fixed Diophantine frequencies and non-cosine analytic potentials, advancing the understanding of localization phenomena.

Findings

Established Anderson localization for a broad class of quasiperiodic operators.

Proved H"older continuity of the integrated density of states.

Introduced a new method for controlling Green's functions and eliminating double resonances.

Abstract

We establish both Anderson localization and H\"older continuity of the integrated density of states for quasiperiodic Schr\"odinger operators on $\mathbb{Z}^d$ with any non-constant analytic potential and any Diophantine frequency in the perturbative regime. Our proof is based on a new method for controlling Green's functions and eliminating double resonances, in the spirit of multi-scale analysis. To the best of our knowledge, this is the first multi-scale analysis approach that works for fixed Diophantine frequencies and potentials beyond the cosine type.