Nonperturbative methods for quasiperiodic discrete Schroedinger equations on the line

Princeton University, Princeton NJ

Investigators

Abstract

ABSTRACT: This proposal deals with various aspects of discrete Schroedinger equations on the one dimensional lattice with deterministic potentials. So far, in collaboration with Jean Bourgain and Michael Goldstein, the author has considered quasi-periodic potentials given by ergodic shifts on tori, potentials obtained by means of the skew-shift on the two torus, as well as potentials defined in terms of strongly mixing dynamics, such as the doubling map on the circle or hyperbolic automorphisms on the two torus. In each of these cases, positivity of the Lyapunov exponent, regularity of the integrated density of states, and Anderson localization were studied. At this point, we are planning to address several remaining questions, including the following ones: 1) Is the Lyapunov exponent positive in case of skew-shift potentials for small disorder ? 2) Is it possible to obtained detailed information on the nature of the eigenfunctions in the quasi-periodic case assuming only positivity of the Lyapunov exponent ? In fact, do the non perturbative techniques allow the definition of the essential support as described in the perturbative regime by Sinai and Froehlich, Spencer, Wittwer ? These questions are intimately linked with Y. G. Sinai's recent work on "anomalous transport in quasi-periodic media", and would provide better and more precise information on the subdiffusive behavior of the random walk considered by Sinai. 3) Is it possible to extend the nonperturbative methods to strips, or the two-dimensional plane ? 4) Is the integrated density of states Holder continuous in the case of several frequencies or the skew-shift ? 5) What can be said about the statistics of the level-spacings of the eigen values for the case of the skew-shift ? Historically, the study of random Schroedinger operators started with Phil Anderson's work in the late 1950's, for which he received the Nobel prize. Before his work it was believed that small random impurities in a crystal would not significantly change its conductance. Anderson, however, showed that this is not the case: Arbitrarily small random impurities occurring independently at each lattice site turn a conductor into an insulator. Since his work, which was not mathematically rigorous, the development of a precise theory of "Anderson localization" has been pursued by many mathematicians. It turned out that there were connections with deep results from several areas of mathematics. For example, Fuerstenberg's theorem on products of random matrices was a crucial tool in the development of the theory. These works attracted the attention of physicists, particularly experts in statistical mechanics. To this day, there is an active and fruitful exchange of ideas between mathematicians and physicists in this subject. In fact, the interest in random phenomena and methods has intensified quite notably in physics in recent years, as many important problems posed by statistical mechanics have proved to be rather deep mathematical challenges whose solution has lead to significant advances of probabilistic techniques. It is our hope that the projects set forth in this proposal will further advance our insight into the models of statistical mechanics as well as providing useful tools for mathematicians working in ergodic theory, analysis, and mathematical physics.