Topics in spectral and inverse spectral theory
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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.