Spectral and Transport Properties of Random Media

This proposal concerns continuing investigations of the principal investigator into the spectral and transport properties of random media. The basic question is: How do random perturbations of a background medium, for example, a perfect crystal, influence the propagation of quantum and classical waves in the medium? The proposed work concentrates on properties of the integrated density of states and models related to the quantum Hall effect. Several other models and related technical problems are discussed. The main aspects of the proposal include: 1) Continuity and Regularity of the Integrated Density of States; 2) Spectral and Transport Problems in the Quantum Hall Effect; 3) the Aizenman-Molchanov Method for Localization, Energy-Level Statistics, and Random Magnetic Fields. The overall goal of this work is to describe the effect of disorder on the measurable properties of the system, such as the density of states, the conductivity, and the edge currents. The proposer also discusses localization for some previously untreated systems like random magnetic fields and nonlocal potentials. The proposed methods will contribute to the understanding of wave propagation in random media. Our understanding of many basic electronic properties of solids is based on the simple one-electron model of an electron propagating in a periodic array of atoms. This simple idea explains many fundamental phenomena such as metals, insulators, and semiconductors. One of the limitations of this model is that it predicts infinite conductivity. In 1958, P. W. Anderson proposed disorder as a mechanism for limiting the ballistic behavior of an electron in a periodic array of atoms. He argued that the electron wave function should be localized in space due to multiple and incoherent scattering from the randomly distributed impurities, with probability one. Disorder-induced localization of states for many models has now been proved. Some of the work in this proposal involves applying these ideas to the study of systems with interesting geometries, such as a strip or a torus, that exhibit the quantum Hall effect. The principal investigator is interested in the nature of the quantum Hall edge-currents, and the nature of the electron states for regions with two-edges, such as a strip. One of the tools for studying these systems is the density of states that provides a measure of the number of electronic states per unit volume. The continuity of the density of states measure provides a measure of the effectiveness of the disorder in breaking degeneracy of the ordered system. Very little is known about this function. The principal investigator is also interested in exploring the effect on electron propagation of randomness in the magnetic field, and the influence of randomness on certain systems with nonlocal interactions.