Analytic quasi-periodic Schrödinger operators and rational frequency approximants

Abstract

Consider a quasi-periodic Schrödinger operator $H_{\alpha ,\theta }$ with analytic potential and irrational frequency $\alpha$. Given any rational approximating $\alpha$, let $S_{+}$ and $S_{-}$ denote the union, respectively, the intersection of the spectra taken over $\theta$. We show that up to sets of zero Lebesgue measure, the absolutely continuous spectrum can be obtained asymptotically from $S_{-}$ of the periodic operators associated with the continued fraction expansion of $\alpha$. This proves a conjecture of Y. Last in the analytic case. Similarly, from the asymptotics of $S_{+}$, one recovers the spectrum of $H_{\alpha ,\theta }.$

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• On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices