All ETDs from UAB

Advisory Committee Chair

Rudi Weikard

Advisory Committee Members

Richard Brown

Yulia Karpeshina

Ryoichi Kawai

Claudio Morales

Document Type

Dissertation

Date of Award

2010

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

ON INVERSE PROBLEMS FOR LEFT-DEFINITE DISCRETE STURM-LIOUVILLE EQUATIONS RAMI ALAHMAD APPLIED MATHEMATICS ABSTRACT We are interested in studying a class of discrete Sturm-Liouville problems called left-de nite problems corresponding to a positive definite quadratic form associated with the left side of the Sturm-Liouville equations. Important tools for studying Sturm-Liouville equations are the eigenfunction expansions and the corresponding Fourier transforms. Here, we introduce the generalized Fourier transform associated with the left-definite problems. Also, we extend the discussion to inverse problems for discrete Sturm-Liouville equations. In inverse spectral problems, the task is to find a coefficient in the equation using the spectral data. We discuss the uniqueness of spectral inverse problems for the discrete left-definite Sturm-Liouville Problems. We introduce the Jost solution corresponding to the left-definite Sturm-Liouville equation to find the coefficients of the left-definite Sturm-Liouville operator by applying the principles of inverse scattering problems. Finally, we find the coefficients of the left-definite Sturm-Liouville operator using the zeros of the Jost solution by applying the principles of inverse resonance problems.

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