Localization, delocalization, and other phenomena in random Schrodinger operators

University Of California-Irvine, Irvine CA

Investigators

Abstract

This project is devoted to the study of localization, delocalization, and other phenomena in random Schrödinger operators, which describe an electron moving in a medium with random impurities. In the widely accepted picture, in three or more dimensions there exists a transition from an insulator region, characterized by localized states, to a very different metallic region, characterized by extended states, while in one or two dimensions there are only localized states and no metal-insulator transition. This project aims to further the mathematical understanding of this picture. The continuum Anderson Hamiltonian with arbitrary single-site probability distribution will be studied, with the objective of proving localization at the bottom of the spectrum, and to characterize the region of dynamical localization by proving a converse to the multiscale analysis, showing existence of a nonzero minimal rate of transport in the complementary region. If single-site probability distribution has a bounded density, a local Wegner estimate and will be proved to obtain Minami's estimate (and hence Poisson statistics for eigenvalues) in the region of localization. The PI will investigate localization in the two-dimensional (discrete) Anderson model by studying the Anderson model on the strip; a transfer matrix approach based on the supersymmetric replica trick will be used. The PI will study a multi-particle Anderson model describing interacting electrons moving in a medium with random impurities, and investigate localization in Fock space. The PI will search for a proof of localization for the Anderson model where the single-site potential is a Bernoulli random variable in two or more dimensions, a known result for the continuum Anderson Hamiltonian. The correct exponent for the logarithmic correction in Mott's formula for the Anderson model will be investigated. The PI will also study Minami's estimate and Poisson statistics for eigenvalues of random classical wave operators (e.g., random acoustic and Maxwell operators), which describe classical waves in random media. Random Schrödinger operators describe an electron moving in a medium with random impurities. In the presence of impurities, a material that normally acts like a metal, i.e., it conducts electric current, will exhibit localization and behave like an insulator for electric currents. The impurities create a metal-insulator transition with important consequences for electric currents. This research will contribute to the understanding of electronic phenomena in condensed matter physics, such as Anderson localization and the quantum Hall effect. Some of the topics of research are suitable for PhD theses, and will be used for the training of future researchers.