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arXiv · 1201.4199

Analytic quasi-periodic Schrödinger operators and rational frequency approximants

S. Jitomirskaya·C. A. Marx
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Abstract

Consider a quasi-periodic Schrödinger operator $H_{α,θ}$ with analytic potential and irrational frequency $α$. Given any rational approximating $α$, let $S_+$ and $S_-$ denote the union, respectively, the intersection of the spectra taken over $θ$. 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 $α$. This proves a conjecture of Y. Last in the analytic case. Similarly, from the asymptotics of $S_+$, one recovers the spectrum of $H_{α,θ}.$

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S. Jitomirskaya, C. A. Marx. 2012-02-13. Analytic quasi-periodic Schrödinger operators and rational frequency approximants. https://arxiv.org/abs/1201.4199

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