Spectral and Transport Properties of Random Media

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Spectral and Transport Properties of Random MediaPI: Peter D. Hislop, University of KentuckyDMS-0202656Abstract:This proposal concerns continuing investigations of the principalinvestigator into the spectral and transport properties of random media.The basic question is: How do random perturbations of a backgroundmedium, for example, a perfect crystal, influence the propagationof quantum and classical waves in the medium?The proposed work concentrates on properties of the integrated density ofstates 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 ofdisorder on the measurable properties of the system,such as the density of states, the conductivity,and the edge currents. The proposer also discusseslocalization for some previously untreated systemslike random magnetic fields and nonlocal potentials.The proposed methods will contribute to theunderstanding of wave propagation in random media.Our understanding of many basic electronic properties of solids isbased on the simple one-electron model of an electron propagatingin a periodic array of atoms. This simple idea explains many fundamentalphenomena such as metals, insulators, and semiconductors. One of thelimitations of this model is that it predicts infinite conductivity. In1958, P. W. Anderson proposed disorder as a mechanism for limiting theballistic behavior of an electron in a periodic array of atoms. He arguedthat the electron wave function should be localized in spacedue to multiple and incoherent scattering from the randomly distributedimpurities, with probability one. Disorder-induced localization of statesfor many models has now been proved. Some of the work in this proposalinvolves applying these ideas to the study of systems with interestinggeometries, such as a strip or a torus, that exhibit the quantum Halleffect. 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 electronicstates per unit volume. The continuity of the density of states measureprovides 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.