Lyapunov Exponents and Spectral Analysis of Ergodic Schr\"odinger Operators: A Survey of Kotani Theory and its Applications

TL;DR

This survey reviews Kotani theory, emphasizing how Lyapunov exponents determine the spectral types of ergodic Schrödinger operators and discusses recent applications in spectral analysis of various potentials.

Contribution

It provides a comprehensive overview of Kotani theory and its recent applications in spectral analysis of ergodic Schrödinger operators.

Findings

Lyapunov exponent fully characterizes absolutely continuous spectrum.

Kotani theory offers tools to prove absence or presence of certain spectral types.

Applications include analysis of rough potentials and the almost Mathieu operator.

Abstract

The absolutely continuous spectrum of an ergodic family of one-dimensional Schr\"odinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolutely continuous spectrum for potentials defined by the doubling map on the circle, and the absence of singular spectrum for the subcritical almost Mathieu operator.