There and Back Again - An Exploration of the Liouville Transformation and Its Inverse

This report summarizes work done as part of the Physics of Strings PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed around the study of a common problem. This module describes a method for recovering the distribution of mass along a one-dimensional non-uniform string from the string's eigenvalues. The eigenvalues are used to construct the string's Sturm-Liouville potential function, and then the mass density is obtained from this potential by means of inverting the Liouville transformation.