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.
Random Schrodinger operators 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. The goal of this proposal was to further the mathematical understanding of localization and delocalization, and related topics. The research outcomes included: A comprehensive proof of localization for continuous Anderson models with singular random potentials, including Bernoulli potentials. This strong form of localization includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) and dynamical localization (no spreading of wave packets under the time evolution). Bounds on the density of states for Schrodinger operators. These are deterministic results that do not require the existence of the integrated density of states. The results are stated in terms of a "density of states outer-measure" that always exists. Log-Holder continuity for the density of states was proved in one, two, and three dimensions for Schrodinger operators, and in any dimension for discrete Schrodinger operators. Unique continuation principle for spectral projections of Schrodinger operators and optimal Wegner estimates for non-ergodic random Schrodinger operators. As an application, optimal Wegner estimates were obtained at all energies for a class of non-ergodic random Schrodinger operators with alloy-type random potentials ('crooked' Anderson Hamiltonians). Optimal Wegner estimates at the bottom of the spectrum were proved with the expected dependence on the disorder. These estimates were applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum. The bootstrap multiscale analysis for the multi-particle Anderson model. The bootstrap multi-scale analysis was extended to the multi-particle Anderson model, obtaining Anderson localization, dynamical localization, and decay of eigenfunction correlations. Ground state energy of trimmed discrete Schrodinger operators and localization for trimmed Anderson models. The restriction of multi-dimensional discrete Schrodinger operators to the complements of given subsets of the lattice were studied, and the dependence of the ground state energy of the excluded subset was determined. It was shown that for relatively dense proper subsets the ground state energy increases. This lifting of the ground state energy was used to establish Wegner estimates and localization at the bottom of the spectrum for trimmed Anderson models, i.e., Anderson models with the random potential supported by the complement of a relatively dense proper subset. Random Schrodinger operators on the Bethe strip. The existence of intervals of purely absolutely continuous spectrum and ballistic behavior for Anderson-like models on the Bethe strip at low disorder was proved. Lifshitz tails estimates for the density of states of the Andersonmodel. An upper bound for the (differentiated) density of states of the Anderson model at the bottom of the spectrum was proved. The density of states was shown to exhibit the same Lifshitz tails upper bound as the integrated density of states.
