Damanik, David T.2019-05-172019-05-172018-052018-04-16May 2018Bucaj, Valmir. "ANDERSON LOCALIZATION FOR DISCRETE ONE-DIMENSIONAL RANDOM OPERATORS." (2018) Diss., Rice University. <a href="https://hdl.handle.net/1911/105722">https://hdl.handle.net/1911/105722</a>.https://hdl.handle.net/1911/105722This thesis is concerned with the phenomenon of Anderson localization for one dimensional discrete Jacobi and Schr\"odinger operators acting on $\ell^2(\Z)$. Specifically, we prove dynamical and spectral localization at all energies for the discrete {\it generalized Anderson model} via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schr\"odinger operators, to the case where a single random variable determines the potential on a block of an arbitrary, but fixed, size $\alpha.$ For this model, we also prove uniform positivity of the Lyapunov exponents. In fact, we prove a stronger statement where we also allow finitely supported distributions. We also show that for any size $\alpha$ {\it generalized Anderson model}, there exists some finitely supported distribution $\nu$ for which the Lyapunov exponent will vanish for at least one energy. Moreover, restricting to the special case $\alpha=1$, we describe a pleasant consequence of this modified technique to the original Kunz-Souillard approach to localization. In particular, we demonstrate that actually the single operator $T_1$ is a strict contraction in $L^2(\mathbb{R})$, whereas before it was only shown that the second iterate of $T_1$ is a strict contraction. This confirms a feature of the original approach that had been expected by experts in the field, but proven to be elusive prior to this thesis. We also study spectral properties of discrete one-dimensional random Jacobi operators. In particular, we prove two main results: (1) that perturbing the diagonal coefficients of Jacobi operators, in an appropriate sense, results in exponential localization, and purely pure point spectrum with exponentially decaying eigenfunctions; and (2) we present examples of decaying {\it potentials} $b_n$ such that the corresponding Jacobi operator has purely pure point spectrum.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Schroedinger OperatorsJacobi OperatorsAnderson LocalizationLyapunov ExponentThesis2019-05-17