All ETDs from UAB

Advisory Committee Chair

Rudi Weikard

Advisory Committee Members

Kenneth B Howell

Ryoichi Kawai

Kabe Moen

Gunter Stolz

Document Type

Dissertation

Date of Award

2013

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

Inverse scattering problems for Sturm-Liouville differential equations find numerous applications in physics, in particular, quantum mechanics. While the theory of these problems has been developed over a number of decades, a more recent concern has been the use of resonances, important phenomena in physics, as data — the inverse resonance problem. In this dissertation, we address this problem in a variety of cases. First, we investigate the full-line Schroedinger equation where the data for the inverse problem include the eigenvalues and resonances. We prove that any two potentials that have enough data points sufficiently close together must also be close in a suitable sense. We then prove a discrete analogue for a full-line Jacobi equation. Finally, we prove a uniqueness theorem for a left-definite, half-line Sturm-Liouville equation. Along the way, we improve upon the current inverse spectral and scattering theorems for this equation.

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