## 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.

## Recommended Citation

Al Ahmad, Rami, "On Inverse Problem for Left-definite Discrete Sturm-Liouville Equations" (2010). *All ETDs from UAB*. 981.

https://digitalcommons.library.uab.edu/etd-collection/981