A solution method to a new class of inverse spectral problems

In this thesis we present a constructive method for the recovery of an unknown potential q(x) in a new family of inverse Sturm-Liouville problems. One of the members of this family is the following:; Let {dollar}{lcub}lambdasb{lcub}n{rcub}{rcub}sbsp{lcub}n=0{rcub}{lcub}infty{rcub}{dollar} be the eigenvalues of the differential equation {dollar}{dollar}-ysp{lcub}primeprime{rcub}+q(x)y=lambda y{dollar}{dollar}subject to the boundary conditions {dollar}{dollar}eqalign{lcub}y(0) - hyspprime(0) &= 0cr y(1) + Hyspprime(1) &= 0cr{rcub}{dollar}{dollar}where the potential {dollar}q(x){dollar} is known to be anti-symmetric about the point {dollar}x={lcub}1over2{rcub}.{dollar} Can we obtain q from {dollar}{lcub}lambdasb{lcub}n{rcub}{rcub}sbsp{lcub}n=0{rcub}{lcub}infty{rcub}?{dollar}; The main idea is to use the spectral data to synthesise the boundary data for a certain Goursat problem and then to use a time domain technique. We prove a uniqueness theorem to the family of inverse problems. We also prove a theorem about the convergence of the method and give some numerical examples.