Tuesday, February 14, 2012

1201.4199 (S. Jitomirskaya et al.)

Analytic quasi-periodic Schrödinger operators and rational frequency

S. Jitomirskaya, C. A. Marx
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 $H_{\alpha,\theta}.$
View original: http://arxiv.org/abs/1201.4199

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