Schr\"odinger Operators with Dynamically Defined Potentials: A Survey

TL;DR

This survey reviews spectral and quantum dynamical properties of one-dimensional Schr"odinger operators with potentials generated by ergodic transformations, emphasizing random and almost periodic cases.

Contribution

It provides a comprehensive overview of the general theory and known results for these operators, highlighting recent advances and specific classes of potentials.

Findings

Spectral properties depend on the nature of the ergodic transformation.

Quantum dynamical behavior varies with potential type.

Known results for random and almost periodic potentials are summarized.

Abstract

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger 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.