Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials

TL;DR

This paper investigates the spectral measure continuity for quasiperiodic Schrödinger operators with rough potentials, establishing that for potentials with Hölder continuity exponent greater than 1/2 and positive Lyapunov exponents, the spectrum measure converges from periodic approximations.

Contribution

It proves the measure of the spectrum for such operators is the limit of spectra of periodic approximants under specified conditions, extending understanding of spectral continuity.

Findings

Spectral measure continuity holds for Hölder potentials with exponent > 1/2.

Spectrum measures of irrational frequency operators are limits of periodic spectra.

Uniformity of the upper Lyapunov exponent is established for strictly ergodic cocycles.

Abstract

We study discrete quasiperiodic Schr\"odinger operators on $\ell^2(\zee)$ with potentials defined by $\gamma$-H\"older functions. We prove a general statement that for $\gamma >1/2$ and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.