Analytic quasi-periodic Schrödinger operators and rational frequency approximants

Details Analytic quasi-periodic Schrödinger operators and rational frequency approximants Analytic quasi-periodic Schrödinger operators and rational frequency approximants

2012-01-20

arxiv.org/abs/1201.4199v2

Physics

Mathematical Physics

Scientific paper

Consider a quasi-periodic Schr\"odinger 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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