Document Type

Theses, Ph.D

Rights

This item is available under a Creative Commons License for non-commercial use only

Publication Details

Successfully submitted for the award of Doctor of Philosophy (Ph.D) to the Dublin Institute of Technology 2005.

Abstract

In this thesis we investigate the properties of a class of linear differential operators known as Sturm-Liouville operators. Sturm-Liouville operators arise from differential equations of the form - y”+q(x)y = λy, where q is known as the potential and λ is a spectral parameter. This differential equation has been of great importance in mathematics and physics alike. We allow the potential to be real or complex valued and assume that q satisfies |q(x)|≤ce-ax, x≥0, where a >0 and c > 0. We first give a brief account of the mathematical background we will be using and then proceed to give a summary of Sturm-Liouville theory on the half-line, following Weyl, Titchmarsh, Sims and Naimark. In chapter three, we investigate the properties of a series derived initially by Eastham, identify zero-free regions for the Jost solution of the Sturm-Liouville equation and conclude by highlighting the significance of our results in terms of the eigenvalues, resonances and spectral singularities of the Sturm-Liouville operator. We also give several examples to support the validity of our results. The last chapter is dedicated to the study of a series derived by Harris and Gilbert. We show that this series can be used to complement the results obtained in the previous chapter and, finally, we use the property of this series to investigate the points of spectral concentration associated with the Sturm-Liouville operator.

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Mathematics Commons

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