The inverse Sturm-Liouville problem: Uniqueness theorems and counterexamples

A number of methods were proposed to generate the uniqueness theorems and counterexamples of the Inverse Sturm-Liouville Problem that is an integrable function. The study of the inverse problem with indecomposable boundary conditions dealt with the reconstruction of the function for the self-adjoint Sturm-Liouville problem with periodic and antiperiodic boundary conditions. A method using certain mappings of spaces of solutions given in matrix form was also applied to study the inverse problems. A set of eigenvalues of the problem, eigenvalues of two auxiliary problems, and additional spectral data were needed to reconstruct the problem. It was found that the algebraic multiplicity of each eigenvalue coincides with that of the corresponding root of the function. The results show that any sufficiently large eigenvalues with numbers of different parity can be used for the inverse problems.