Spectral and Transport Theory of Schrodinger Operators

University Of California-Irvine, Irvine CA

Investigators

Abstract

This research involves localization type effects for ergodic Schrodinger operators, the general study of spectra and wave functions of multidimensional Schrodinger operators, and also of the transport phenomena of quasiperiodic operators and two-dimensional magnetic operators in the integer quantum Hall regime. An important objective is to develop nonperturbative methods of proving localization type effects for Schrodinger operators with deterministic potentials. The other goal is to study the relation between spectral and quantum-dynamical properties and behavior of the generalized eigenfunctions, particularly outside the localization range and in the multidimensional case. Another goal of the proposed research is to study singular continuous spectrum that exhibits critical behavior and/or anomalous transport, particularly for models where it appears for critical values (or intervals) of the parameters. The proposed research is centered around the fundamental properties of disordered systems that serve as models of systems with impurities. Deterministic, particularly quasiperiodic, potentials are most often used to model quasicrystals. In order to be able to understand much of the experimental data on quasicrystals, it is particularly important to investigate the transport coefficients like the electrical and heat conductivities of the microscopic models. Such an understanding is most helpful for finding new materials with desired physical properties. This may lead to various industrial applications (the first one nowadays being the covering of pans replacing the conventional Tefal film). Disordered systems are also used in modeling many other micro and macro effects: from quantum localization to earthquakes. Our research concerns the anomalous spectral and diffusive properties of quasiperiodic and other deterministic structures. The quantum Hall effect is since 1985 used by the National Bureau of Standards to define the Fine Structure Constant (and hence the electrical charge of an electron). It is still not well understood why the experiment can be reproduced with a relative error of only $10^{-8}$. Our research is concerned with a microscopic theory of the quantum Hall effect that is aimed at getting deeper insights of this phenomena.