The first of Liouville’s theorems in this monograph is concerned with the linear eigenvalue problem in mathematical physics. Having a summary in the first section, we begin with the applied inverse problem in the second section, to introduce boundary identification of parabolic equations and its ill-posedness to reach the inverse Sturm–Liouville problem. In the third section, the Gel’fand–Levitan– Marchenko (GLM) equation is used to reduce the problem to a study on Goursat problems for hyperbolic equations. We are thus lead to eigenvalue problems in the fourth section, and the intersection property as a striking tool in the theory for one space dimension in the fifth section. We then come back to the inverse Sturm–Liouville problem in the sixth section; uniqueness, reconstruction, and linearization of the Hochstadt equation. The \(\tau \) -function emerges as a Fredholm determinant in the final section, from symmetry of the problem and duality between the boundary conditions; those of Dirichlet and Robin.