An Introduction to the Mathematics of Anderson Localization

TL;DR

This paper provides a comprehensive mathematical introduction to Anderson localization, covering spectral and dynamical localization, the fractional moments method, and open problems in the field.

Contribution

It offers a detailed, self-contained explanation of the Anderson model and extends the fractional moments method to the continuum case, highlighting key open problems.

Findings

Proves localization properties of the Anderson model

Discusses spectral and dynamical localization

Extends fractional moments method to continuum models

Abstract

We give a widely self-contained introduction to the mathematical theory of the Anderson model. After defining the Anderson model and determining its almost sure spectrum, we prove localization properties of the model. Here we discuss spectral as well as dynamical localization and provide proofs based on the fractional moments (or Aizenman-Molchanov) method. We also discuss, in less self-contained form, the extension of the fractional moment method to the continuum Anderson model. Finally, we mention major open problems. These notes are based on several lecture series which the author gave at the Kochi School on Random Schr\"odinger Operators, November 26-28, 2009, the Arizona School of Analysis and Applications, March 15-19, 2010 and the Summer School on Mathematical Physics, Sogang University, July 20-23, 2010.