Continuity of the measure of the spectrum for discrete quasiperiodic operators

Abstract

We study discrete Schroedinger operators $(H_{\alpha,\theta}\psi)(n)= \psi(n-1)+\psi(n+1)+f(\alpha n+\theta)\psi(n)$ on $l^2(Z)$, where $f(x)$ is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of $H_{\alpha,\theta}$ to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of $H_{\alpha,\theta}$ are positive. For the almost Mathieu operator ($f(x)=2\lambda\cos 2\pi x$) it follows that the measure of the spectrum is equal to $4|1-|\lambda||$ for all real $\theta$, $\lambda\ne\pm 1$, and all irrational $\alpha$.