Dynamics and spectral theory of quasi-periodic Schrödinger-type operators

Abstract

Quasi-periodic Schrödinger-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 Schrödinger-type operators attaining this "global" view-point with an emphasis on dynamical aspects of the spectral theory of such operators.

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Cited by in corpus (33)

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• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Anderson localization for multi-frequency quasi-periodic operators on ${\mathbb{Z}}^d$
• Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase
• Singular continuous Cantor spectrum for magnetic quantum walks
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• Spectral decimation of a self-similar version of almost Mathieu-type operators
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• On the Spectrum of the Periodic Anderson-Bernoulli Model
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• Schrödinger Operators with Thin Spectra
• Almost Mathieu operators with completely resonant phases
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• Density of states and Delocalization for discrete magnetic random Schrödinger operators
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