Avila's acceleration via zeros of determinants, and applications to Schr\"odinger cocycles

TL;DR

This paper characterizes Avila's quantized acceleration of Lyapunov exponents through zeros of determinants and applies this to prove regularity and localization results for quasi-periodic Schrödinger operators.

Contribution

It introduces a new characterization of Avila's acceleration via zeros of Dirichlet determinants and applies it to establish continuity and localization properties in spectral theory.

Findings

Proves Hölder continuity of the integrated density of states in certain regimes.

Establishes Anderson localization for Diophantine frequencies with analytic potentials.

Provides a new perspective on Lyapunov exponents via zeros of determinants.

Abstract

In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove $\beta$-H\"older continuity of the integrated density of states for supercritical quasi-periodic Schr\"odinger operators restricted to the $\ell$-th stratum, for any $\beta<(2(\ell-1))^{-1}$ and $\ell\ge2$. We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.