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

TL;DR

This paper explores the relationship between the spectra of almost periodic and periodic discrete Schrödinger operators with analytic potentials, providing bounds and new proofs related to their spectral properties.

Contribution

It establishes a weak connection between the absolutely continuous spectrum in the almost periodic case and the spectra in the periodic case, and generalizes Chambers' formula for new insights.

Findings

Bound the measure of the spectrum in the periodic case using Lyapunov exponents

Prove a weak form of a conjecture linking almost periodic and periodic spectra

Provide a new proof of Herman's lower bound for the Lyapunov exponent

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.