Spectral Properties of Limit-Periodic Schr\"odinger Operators (PhD Thesis)

TL;DR

This thesis explores the spectral diversity of limit-periodic Schrödinger operators, demonstrating that all spectral types can occur and providing examples of almost periodic operators with uniform localization.

Contribution

It introduces a new perspective on limit-periodic potentials via minimal translations of procyclic groups, enabling a comprehensive spectral classification and localization results.

Findings

All spectral types can occur in limit-periodic Schrödinger operators.

First examples of uniformly localized almost periodic operators across the spectrum.

Analysis of Lyapunov exponents related to spectral types.

Abstract

We investigate spectral properties of limit-periodic Schr\"odinger operators in $\ell^2(\Z)$. Our goal is to exhibit as rich a spectral picture as possible. We regard limit-periodic potentials as generated by continuous sampling along the orbits of a minimal translation of a procyclic group. This perspective was first proposed by Avila and further exploited by the author, which allows one to separate the base dynamics and the sampling function. Starting from this point of view, we conclude that all the spectral types (i.e. purely absolutely continuous, purely singular continuous, and pure point) can appear within the class of limit-periodic Schr\"odinger operators. We furthermore answer questions regarding how often a certain type of spectrum would occur and discuss the corresponding Lyapunov exponent. In the regime of pure point spectrum, we exhibit the first almost periodic examples that are uniformly localized across the hull and the spectrum.