Spectral Properties of Limit-Periodic Operators

TL;DR

This survey explores the spectral characteristics of limit-periodic operators, highlighting the diversity of spectral types, the prevalence of Cantor spectra, and the range of regularity in the integrated density of states across various operator classes.

Contribution

It provides a comprehensive overview of spectral properties of limit-periodic operators, including proofs and examples across multiple operator classes, emphasizing the diversity and richness of spectral phenomena.

Findings

All basic spectral types occur for dense sets of potentials.

The spectrum often forms a Cantor set, but can also be gapless.

The regularity of the integrated density of states varies widely.

Abstract

We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum Schr\"odinger operators and multi-dimensional Schr\"odinger operators, are discussed as well. We explain that each basic spectral type occurs, and it does so for a dense set of limit-periodic potentials. The spectrum has a strong tendency to be a Cantor set, but there are also cases where the spectrum has no gaps at all. The possible regularity properties of the integrated density of states range from extremely irregular to extremely regular. Additionally, we present background about periodic Schr\"odinger operators and almost-periodic sequences. In many cases we outline the proofs of the results we present.