Schrödinger Operators with Dynamically Defined Potentials: A Survey

Abstract

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

81 pages, to appear in Ergodic Theory Dynam. Systems

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• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• The absolutely continuous spectrum of the almost Mathieu operator
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• Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems
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• A family of Schrödinger operators whose spectrum is an interval
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