2:00 pm Wednesday, April 4, 2018
Thesis Defense: Anderson Localization for Discrete One-Dimensional Random Operators
by Valmir Bucaj, Ph.D. Candidate (Rice University) in Sewall Hall 305
This thesis is concerned with the phenomenon of Anderson localization for one dimensional discrete Jacobi and Schroedinger operators acting on $\ell^2(\Z)$. Specifically, we prove dynamical and spectral localization at all energies for the discrete generalized Anderson model via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schroedinger operators, to the case where a single random variable determines the potential on a block of an arbitrary, but fixed, size N. For this model, we also prove uniform positivity of the Lyapunov exponent. In fact, we prove a stronger statement where we also allow finitely supported distributions. We also show that for the generalized Anderson model, of any block size, there exists some finitely supported distribution $\nu$ for which the Lyapunov exponent will vanish for at least one energy. We also show that any Jacobi operator with bounded coefficients can be approximated, in operator norm, by Jacobi operators with slow-enough decaying diagonal entries and pure point spectrum. Host Department: Rice University-Mathematics
Submitted by lpl1@rice.edu |
