Continuity of the measure of the spectrum for discrete quasiperiodic operators

2001/07/31 by S. Ya. Jitomirskaya, I. V. Krasovsky
Mathematics · Physics and Astronomy · #math.SP #math-ph #math.MP

paper · pdf

published as Math.Res.Lett. 9 (2002) 413-422 · 10 pages, small changes, to appear in Math.Res.Lett

arxiv created 2002/05/30 · arxiv updated 2009/11/30

Abstract

We study discrete Schroedinger operators (H_α,θψ)(n)= ψ(n-1)+ψ(n+1)+f(αn+θ)ψ(n) on l²(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_α,θ to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of H_α,θ are positive. For the almost Mathieu operator (f(x)=2λcos 2πx) it follows that the measure of the spectrum is equal to 4|1-|λ|| for all real θ, λ≠± 1, and all irrational α.

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