Browsing by Author "Damanik, David T."

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    ANDERSON LOCALIZATION FOR DISCRETE ONE-DIMENSIONAL RANDOM OPERATORS
    (2018-04-16) Bucaj, Valmir; Damanik, David T.
    This 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.
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    Transport Properties of One-dimensional Quantum Systems
    (2018-08-08) Gerbuz, Vitalii; Damanik, David T.
    We study quantum dynamical properties of discrete one-dimensional models of Schrodinger operators. The first goal of this thesis is to understand the time evolution of initial states supported on more than one site. We develop tools to bound the so-called transport exponents both from below and from above and apply them to several models. In particular we extend a group of results concerning Sturmain Hamiltonians, quasi-periodic Hamiltonians, substitution generated models and random polymer model. The second topic is a detailed analysis of the transport exponents in the case of a Sturmain model when the frequency is a quadratic irrational. In this case methods from hyperbolic dynamics are applied to study the trace map of the operator. We show that the wavepacket spreads out with the same polynomial rate on all possible timescales. The last model of interest is the Anderson model. We provide a new proof of the Anderson localization phenomenon and derive dynamical localization bounds for states supported on more than one site. This thesis contains joint work with Valmir Bucaj, David Damanik, Jake Fillman, Tom VandenBoom, Fengpeng Wang, and Zhenghe Zhang.