Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues

TL;DR

This paper develops non-perturbative methods to analyze the integrated density of states (IDS) in almost Mathieu models, establishing its regularity, stability, and eigenvalue separation properties under certain conditions.

Contribution

It introduces new techniques for studying the IDS's regularity and eigenvalue gaps, including a multiscale approach and a mechanism for eliminating resonant phases.

Findings

IDS has almost 1/2k Hölder continuity under positive Lyapunov exponents

IDS is Lipschitz outside a zero Hausdorff dimension set

IDS is absolutely continuous for almost every shift angle

Abstract

We develop some non-perturbative methods for studying the IDS in almost Mathieu and related models. Assuming positive Lyapunov exponents, and assuming that the potential function is a trigonometric polynomial of degree k, we show that the Holder exponent of the IDS is almost 1/2k. We also show that this is stable under small perturbations of the potential (e.g., potentials which are close to that of almost Mathieu again give rise to almost 1/2 Holder continuous IDS). Moreover, off a set of Hausdorff dimension zero the IDS is Lipschitz. We further deduce from these properties that the IDS is absolutely continuous for almost every shift angle. The proof is based on large deviation theorems for the entries of the transfer matrices in combination with the avalanche principle for non-unimodular matrices. These two are combined to yield a multiscale approach to counting zeros of determinants which fall into small disks. In addition, we develop a new mechanism of eliminating resonant phases, and use it to obtain lower bounds on the gap between Dirichlet eigenvalues. The latter also uses the exponential localization of the eigenfunctions.