Topics in spectral and inverse spectral theory

Abstract

This dissertation is concerned with two major classes of operators and provides various spectral and inverse spectral results for them. In the first part of this work a special class of one-dimensional discrete unitary operators is under investigation. The underlying Weyl-Titchmarsh theory and a Borg-type inverse spectral result are established for this class of operators. The second part of this work is devoted to some spectral theoretical questions for one- and multi-dimensional Schrödinger operators. In particular, the Weyl-Titchmarsh theory for one-dimensional self-adjoint Schrödinger operators with strongly singular potentials is established. In addition, a general perturbation theory for non-self-adjoint operators is developed and subsequently applied to a large class of non-self-adjoint multi-dimensional Schrödinger operators.