Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 509

Continuity of the measure of the spectrum for discrete quasiperiodic operators

S. Ya. Jitomirskaya, I. V. Krasovsky

We study discrete Schr"odinger operators $(H_{alpha, heta}psi)(n)= psi(n-1)+psi(n+1)+f(alpha n+ heta)psi(n)$ 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 $H_{alpha, heta}$ to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of $H_{alpha, heta}$ are positive. For the almost Mathieu operator ($f(x)=2lambdacos 2pi x$) it follows that the measure of the spectrum is equal to $4|1-|lambda||$ for all real $ heta$, $lambda epm 1$, and all irrational $alpha$.

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