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 $\alpha$.

10 pages, small changes, to appear in Math.Res.Lett