Continuity of the measure of the spectrum for discrete quasiperiodic operators

Details Continuity of the measure of the spectrum for discrete quasiperiodic operators Continuity of the measure of the spectrum for discrete quasiperiodic operators

2001-07-08

arxiv.org/abs/math/0107061v2

Math.Res.Lett. 9 (2002) 413-422

Mathematics

Spectral Theory

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

Scientific paper

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 $\alpha$.

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