Lifschitz Tails for Random Schr\"{o}dinger Operator in Bernoulli Distributed Potentials

This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schr\"{o}dinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite

Box approximation and related techniques in spectral theory

The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file.Title from title screen of research.pdf file (viewed on June 2, 2009)Vita.Includes bibliographical references.Thesis (Ph. D.) University of Missouri-Columbia 2008.Dissertations, Academic -- University of Missouri--Columbia -- Mathematics.This dissertation is concerned with various aspects of the spectral theory of differential and pseudodifferential operators. It consists of two chapters. The first chapter presents a study of a family of spectral shift functions [xi]r, each associated with a pair of self-adjoint Schrödinger operators on a finite interval (0, r). Specifically, we investigate the limit behavior of the functions [xi]r when the parameter r approaches infinity. We prove that an ergodic limit of [xi]r coincides with the spectral shift function associated with the singular problem on the semi-infinite interval. In the second chapter, we study the attractor of the dynamical system r [arrow] Ar, where Ar is the truncated Wiener-Hopf operator surrounded by operators of multiplication by the function e[superscript alpha/2] [absolute value of dot], [alpha][greater than] 0. We show that in the case when the symbol of the Wiener-Hopf operator is a rational function with two real zeros the dynamical system r [arrow] Ar possesses a nontrivial attractor of a limit-circle type

Dispersive Estimates for Harmonic Oscillator Systems

We consider a large class of harmonic systems, each defined as a quasi-free dynamics on the Weyl algebra over $\ell^2(\mathbb{Z}^d)$. In contrast to recently obtained, short-time locality estimates, known as Lieb-Robinson bounds, we prove a number of long-time dispersive estimates for these models

Lifschitz Tails for Random Schr\"{o}dinger Operator in Bernoulli Distributed Potentials

This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schr\"{o}dinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite