The matrix form of sturm-liouville problems with finite spectrum

This thesis based on the fact that for every integer n, we construct a class of regular self-adjoint and non-self-adjoint Sturm-Lioville problems with axactly n eigenvalues. In this thesis, first we define a class of Sturm-Liouville equations will be named with Atkinson equation such that any Sturm-Lioville problem consisting of such an equation and an arbitrary seperated and coupled real self-adjoint boundary condition has a represantation as an equivalent finite dimension matrix eigenvalue problem. Then conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or coupled real self-adjoint boundary condition, we construct a class of Sturm–Liouville problems with the specified boundary condition, each of which is equivalent to the given matrix eigenvalue problem. Equivalent here means that they have exactly the same eigenvalues. We considered two type Sturm-Lioville problem which are regular Sturm-Lioville problem and Sturm-Lioville problem with transmission condition.