Spectral Theory of Random Schroedinger Operators and Matrices

Spectral Theory of Random Operators and Matrices The PI is engaged in the study of the local eigenvalue statistics (LES) for models of disordered systems focusing on random Schroedinger operators and random band matrices. These are models of the propagation of electrons in randomly disordered media. Transitions in the LES are expected to reflect the transport transition between localization and delocalization of quantum states. For the localization regime, recent results with W. Kirsch and M. Krishna involve proof of the Poisson nature of local eigenvalue statistics for delta-interaction models in dimensions less than or equal to 3, the compound Poisson nature of the LES for higher-rank Anderson models, and the general nature of the LES as infinitely-divisible point processes with Levy measures supported on the natural numbers. In related work, the PI and C. Marx are studying the dependence of the density of states on the potential for ergodic Schroedinger operators. Our novel approach uses the concept of the density of states outer measure (DOSoM) introduced by Bourgain and Klein for any Schroedinger operator. Results include the modulus of continuity of the DOSoM and associated distribution function. These deterministic result provide sharp bounds when restricted to ergodic models. Among related results is the modulus of continuity of the Lyapunov exponent for one-dimensional models with respect to the potential and in the weak disorder regime. This is the basis of current works to extend these results of the continuity of the Lyapunov exponent to the random Schroedinger operator on the Bethe lattice. The expected continuity of the Lyapunov exponent in the disorder will provide a tool necessary for the completion of the description of the phase diagram of a random Schroedinger operator on the Bethe lattice. Recent work on random band matrices (RBM) with former student B. Brodie establish the Poisson nature of the LES for random band matrices with fixed bandwidth and the smoothness of the density of states. The PI and M. Krishna are completing extensions of these results for RBM with bandwidths growing more slowly than the conjectured transition rate of the square root of the matrix size. These are the first results on LES for RBM in the localization region.