Anderson localization for the discrete one-dimensional quasi-periodic Schroedinger operator with potential defined by a Gevrey-class function

TL;DR

This paper proves Anderson localization for a class of one-dimensional quasi-periodic Schrödinger operators with Gevrey-class potentials, showing positive Lyapunov exponents and continuity properties under certain conditions.

Contribution

It establishes Anderson localization and related spectral properties for Gevrey-class potentials in the perturbative regime, extending previous results to a broader class of functions.

Findings

Anderson localization holds for large disorder and most frequencies.

Lyapunov exponent is positive for all energies.

Lyapunov exponent and density of states are continuous with a specific modulus.

Abstract

In this paper we consider the discrete one-dimensional Schroedinger operator with quasi-periodic potential v_n = \lambda v (x + n \omega). We assume that the frequency \omega satisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove - in the perturbative regime - that for large disorder \lambda and for most frequencies \omega the operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes.