Finite spectrum of Sturm-Liouville problems with delta interactions and matrix representations

The goal of this thesis is to study the finite spectrum and their matrix representations of Sturm-Liouville problems with delta interactions. Indeed, such a problem is equivalent to a Sturm-Liouville problem which has n transmission conditions. Firstly, we show that we can construct a Sturm-Liouville problem with delta interactions, which has exactly at most d eigenvalues. Where d is the sum of positive numbers that are related to number of the partition of the intervals between two successive interaction points. Secondly, we also show that the Sturm-Liouville problem with delta interactions is equivalent to a finite dimensional matrix eigenvalue problem.