Spectral Properties of Schr\"odinger Operators Arising in the Study of Quasicrystals

TL;DR

This survey reviews spectral properties of Schr"odinger operators linked to quasicrystals, emphasizing one-dimensional models like the Fibonacci Hamiltonian, recent methods from hyperbolic dynamics, and numerical insights supporting theoretical results.

Contribution

It provides a comprehensive overview of spectral results for quasicrystal models, highlighting recent methods and detailed analysis of key examples like the Fibonacci Hamiltonian.

Findings

Spectral properties of Fibonacci Hamiltonian are well-understood.

Hyperbolic dynamics methods have advanced the study of Sturmian potentials.

Numerical calculations support and suggest new conjectures for spectral analysis.

Abstract

We survey results that have been obtained for self-adjoint operators, and especially Schr\"odinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the one-dimensional case, and in particular on several key examples. The most prominent of these is the Fibonacci Hamiltonian, for which much is known by now and to which an entire section is devoted here. Other examples that are discussed in detail are given by the more general class of Schr\"odinger operators with Sturmian potentials. We put some emphasis on the methods that have been introduced quite recently in the study of these operators, many of them coming from hyperbolic dynamics. We conclude with a multitude of numerical calculations that illustrate the validity of the known rigorous results and suggest conjectures for further exploration.