The inverse spectral problem of fourth-order transmission conditions sturm-liouville problems

The aim of this thesis is based on the creation of a class of self-adjoint and non-self- adjoint fourth-order Sturm-Liouville problems with a transmission condition finite spectrum. In this thesis, we will define a type of Sturm-Liouville problem that will later be called the Atkinson equation. The fourth-order Sturm-Liouville inverse spectral problem, which is formed by this kind of equation with seperated coupled self- adjoint boundary conditions, has a finite dimensional matrix eigenvalue problem. Conversely, for any given type of matrix eigenvalue problem and an arbitrary seperated coupled self- adjoint boundary condition, we will construct a class of Sturm-Liouville problems with a special boundary condition equivalent to this matrix eigenvalue problem. Equivalence here means having the same eigenvalue. Key words: Sturm-Liouville Problem, Finite Spectrum, Matrix Eigenvalue Problems, Inverse Spectral Problem