Some connections between almost periodic and periodic discrete Schroedinger operators with analytic potentials

2011/01/25 by Mira Shamis, Shamis, Mira
Computer Science · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1101.4700

openalex publication_date 2011/01/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study discrete Schroedinger operators with analytic potentials. In particular, we are interested in the connection between the absolutely continuous spectrum in the almost periodic case and the spectra in the periodic case. We prove a weak form of a precise conjecture relating the two. We also bound the measure of the spectrum in the periodic case in terms of the Lyapunov exponent in the almost periodic case. In the proofs, we use a partial generalization of Chambers' formula. As an additional application of this generalization, we provide a new proof of Herman's lower bound for the Lyapunov exponent.

Citations

• Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity
• A formula with some applications to the theory of Lyapunov exponents
• Products of Random Matrices

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