10.5 There and back again - an exploration of the liouville

Introduction

Spectral analysis – the study of the eigenvectors and eigenvalues of linear operators between vector spaces – is one of the most important areas ofresearch in modern applied mathematics, with applications in areas as diverse as mechanics, signal processing, and biology. The spectral analysis ofoperators involved in partial differential equations is especially important, as it offers keen insight into some of the fundamental laws that make modernengineering possible. Of all these equations, the wave equation in a single dimension is perhaps the simplest and easiest to understand, yet the mathematicsthat underlie it is both rich and beautiful. In particular, it provides an illuminating platform from which to study spectral theory, for the eigenvaluesof 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 questionis specified, and one is asked to determine its eigenvalues and eigenvectors. In inverse problems, one is instead given information about an operator'seigenstructure and is asked to reconstruct the operator. In the context of the vibrating string problem, the forward problem asks one to find a string'sfrequencies of vibration given its physical characteristics, while the inverse problem seeks knowledge of the string's physical properties from itsfrequencies. In the case of the vibrating string, both of these problems have been well studied, and various techniques exist for solving them for stringswith 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 spectralcharacteristics are known. In the case of the one-dimensional wave equation [link] , it is possible to transform it into the Sturm-Liouville equation [link] , whose spectral properties have been well-studied (see, e.g., [link] ). While such transformations are mathematically very convenient, the original physicalinterpretation of the problem becomes lost amid changes-of-variables. That is, in order to make physical sense of the solution one must ”untransform" itback into original problem's domain. The focus of our work and of this paper is on this inversion process, specifically for the transformation between theone-dimensional wave equation and its Sturm-Liouville counterpart.