Inverse spectral problems with incomplete knowledge of the spectrum

To solve an inverse spectral problem, we try to discover an operator of a certain form that has a prescribed spectrum. In this thesis we proceed in two different settings, both times considering the potential function as the unknown part of the operator. In the first case, we consider symmetric matrices where the diagonal matrices are the potential functions to be found when we are required to produce a given partial list of eigenvalues. We begin with an investigation showing that given sufficiently many control parameters for our diagonal matrices, we may solve the inverse problem for an open dense set of matrices. Then, following the work of K. P. Hadeler, we turn our attention to discovering a condition on a partial list of target eigenvalues so that we may employ a fixed point theorem to prove the existence of the required diagonal matrix. In the Sturm-Liouville setting, we restrict our potential function, V(x), to a space of step-functions with a finite number of coefficients controlling the heights of the steps. We find a condition on the list of target eigenvalues so that the list appears as a subset of the spectrum of the operator -d2dx2+V x acting on functions with Dirichlet boundary conditions on the interval [0, 1]. By using domain monotonicity and a standard comparison theorem for operators, we give an inductive argument proving the existence of the step-function coefficients. The key idea is to produce eigenvalues on the subintervals which approximate the eigenvalues on the whole interval, then demonstrating that the exact solution exists