Analytic quasi-periodic Schrödinger operators and rational frequency approximants

2012/01/20 by Svetlana Jitomirskaya, Jitomirskaya, S., Christoph A. Marx +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1201.4199

openalex publication_date 2012/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Almost reducibility and absolute continuity I

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