Sturm-liouville problems with finite spectrum

A Standard and well-known result in Sturm-Liouville theory states that the spectrum of a regular or singular, self-adjoint Sturm-Liouville Problem (SLP) is unbounded and therefore infinite. In this thesis, we construct a self-adjoint and non-self adjoint SLP's with exactly n eigenvalues for each non-negative integer n. We also show that these n eigenvalues can be arbitrarily distributed throughout the complex plane C or the real axis R. Then this problem is extended to Sturm-Liouville problems with transmission condition and with eigenparameter dependent boundary conditions.