Analytic quasi-periodic Schr\"odinger operators and rational frequency approximants

TL;DR

This paper proves a conjecture relating the spectrum of quasi-periodic Schrödinger operators with irrational frequency to the spectra of their rational approximants, revealing how the absolutely continuous spectrum can be asymptotically derived.

Contribution

It establishes a link between the spectrum of irrational-frequency operators and their rational approximants, confirming a conjecture of Y. Last in the analytic setting.

Findings

The absolutely continuous spectrum can be asymptotically obtained from the spectra of periodic operators.

The spectrum of the quasi-periodic operator can be recovered from the asymptotics of rational approximants.

The results confirm a conjecture of Y. Last for the analytic case.

Abstract

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 $H_{\alpha,\theta}.$