2/16/2005

Michael Goldstein
University of Toronto

Anderson localization: the state of the problem and applications

Eigen-functions and spectrum of Schrödinger equation with potentials exhibiting random behavior were studied extensively in the last forty years starting from the famous works of Anderson and Harper. Properties of this type of equations are fundamental in understanding phase transitions in quantum mechanical disordered systems of solid state physics. Besides their relevance to physics these equations suggest a rich mathematical program. The central part of this program consists of the study of the structure of the so-called set of resonances and intersections of the different shifts of this set under the translations in the space of potentials. The most important questions regarding the properties of the eigen-functions, in particular their exponential decay, known as Anderson localization, are closely related to this set. The answers to these questions are expected to depend on the dimension of the problem and also on the stochastic properties of the translations in the space of potentials (regular stationary processes, hyperbolic dynamical systems and just i.i.d. random values like in the Anderson model or quasi-periodic dynamics like in Harper’s model). These questions were studied first in perturbative regimes with use of ideas of KAM theory. In the last five years new methods of the analysis of resonances for quasi-periodic and skew-shifted dynamics were developed in the works of Bourgain, Goldstein and Schlag. In this talk we will describe the status of the main problems in this field, some recent results and prospective applications.