 01256 S. Ya. Jitomirskaya, I. V. Krasovsky
 Continuity of the measure of the spectrum for discrete
quasiperiodic operators
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Jul 10, 01

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Abstract. We study discrete Schr\"odinger operators
$(H_{\alpha,\theta}\psi)(n)=
\psi(n1)+\psi(n+1)+f(\alpha n+\theta)\psi(n)$ 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 $41\lambda$ for
all real $\theta$, $\lambda\ne\pm 1$, and all irrational $\alpha$.
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