Spectral analysis – the study of the eigenvectors and eigenvalues of linear operators between vector spaces – is one of the most important areas of research in modern applied mathematics, with applications in areas as diverse as mechanics, signal processing, and biology. The spectral analysis of operators involved in partial differential equations is especially important, as it offers keen insight into some of the fundamental laws that make modern engineering possible. Of all these equations, the wave equation in a single dimension is perhaps the simplest and easiest to understand, yet the mathematics that underlie it is both rich and beautiful. In particular, it provides an illuminating platform from which to study spectral theory, for the eigenvalues of the underlying differential operator bear a straightforward physical interpretation as the vibrational frequencies of the string being modeled.

Within spectral theory, there are two broad classes of problems: forward problems and inverse problems. In forward problems, the operator in question is specified, and one is asked to determine its eigenvalues and eigenvectors. In inverse problems, one is instead given information about an operator's eigenstructure and is asked to reconstruct the operator. In the context of the vibrating string problem, the forward problem asks one to find a string's frequencies of vibration given its physical characteristics, while the inverse problem seeks knowledge of the string's physical properties from its frequencies. In the case of the vibrating string, both of these problems have been well studied, and various techniques exist for solving them for strings with both continuous and discrete mass densities.

One possible approach to the spectral analysis of a partial differential equation is to transform it into another form for which the spectral characteristics are known. In the case of the one-dimensional wave equation Equation 1, it is possible to transform it into the Sturm-Liouville equation Equation 6, whose spectral properties have been well-studied (see, e.g., [3]). While such transformations are mathematically very convenient, the original physical interpretation of the problem becomes lost amid changes-of-variables. That is, in order to make physical sense of the solution one must ”untransform" it back into original problem's domain. The focus of our work and of this paper is on this inversion process, specifically for the transformation between the one-dimensional wave equation and its Sturm-Liouville counterpart.

The remainder of this document is divided as follows. In "Problem Setup", we provide a short introduction to the forward and inverse problems under consideration. In "The Transformations", we analyze in detail the transformation used to turn the wave equation into the Sturm-Liouville equation and describe a method for inverting it. In "Numerical Results", we present the results of the main collection of numerical experiments we conducted, which use a particular Sturm-Liouville potential function from [3]. A discussion of the errors in these results is presented in "Error Analysis: Recovering Eigenvalues". "Corresponding q(t) and ρ(x) Functions in Closed Form" contains results for other functions in closed form we studied. Finally, we provide a brief description of the numerical methods we used in "Numerical Methods".