Dynamics and spectral theory of quasi-periodic Schr\"odinger-type operators

TL;DR

This paper surveys the spectral and dynamical properties of quasi-periodic Schr"odinger-type operators, highlighting recent model-independent approaches and phenomena like criticality and singular continuous spectrum in these systems.

Contribution

It provides a comprehensive overview of the global spectral theory of quasi-periodic Schr"odinger operators, emphasizing dynamical methods and phenomena beyond specific models.

Findings

Identification of critical phenomena with singular continuous spectrum

Application of dynamical systems techniques to spectral analysis

Evidence of model-independent spectral features

Abstract

Quasi-periodic Schr\"odinger-type operators naturally arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics (several of B. Simon's problems) and physics (e.g. Graphene, quantum Hall effect). Whereas research in the first three decades was focused mainly on unraveling the unusual properties of the AMO and operators with similar structure of potential, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," model-independent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of criticality corresponding to purely singular continuous spectrum for a measure theoretically typical realization of the phase, could be tested for prevalence in general models. The intention of this article is to survey the theory of quasi-periodic Schr\"odinger-type operators attaining this "global" view-point with an emphasis on dynamical aspects of the spectral theory of such operators.