Spectral problems for the matrix sturm-liouville equations with the potential depending on the complex parameter

Because of many applications in the fields of physics and engineering, the Sturm-Liouville operators have an important place in the spectral theory of differential operators and are always in the interest of scientists. Though centuries have passed, this theory continues to evolve with the emergence of new problems. Specially, in the middle of the last century, after inverse problems of spectral analysis were used in solution of some nonlinear mathematical physics equations, also called evolution equations, the interest in the Sturm-Liouville theory has increased. In the spectral theory of Sturm-Liouville operators, inverse problems are problems of determining a linear operator according to spectral characteristics such as spectrum, spectral function, scattering data. First results on inverse problems for the classical Sturm-Liouville operator were obtained by mathematicians like G. Borg, N. Levinson, B. M. Levitan, I.M. Gelfand, M.G. Gasymov, V.A. Marchenko, L.D. Faddeev and It has become an advanced theory today. The inverse problem theory for the matrix Sturm-Liouvile equation and for the system of Sturm-Liouville equations in the semi axis and entire real line has been also started by works of Z.S.Agranovich and developed parallel to the classical situation. However, there are very few studies on inverse problems for the matrix Sturm-Liouville operators on the finite interval. Specially, when the potential function is depended on the spectral parameter, the inverse problems related with the matrix Sturm-Liouville pencils are not almost studied and there are many problems that remain open when compared to the scalar case.. This master thesis proposal deals with spectral problems related to separated boundary conditions for the second order and fourth order pencils of the matrix Sturm-Liouville equation. First of all, we purpose to investigate the direct problem which consists of studying of spectral properties of the Sturm-Liouville boundary problem and some asymptotic behaviors for eigenvalues and eigenfunctions. After these, the inverse problems , namely the determination of the coefficients of the matrix Sturm-Liouville equation with respect to the spectral data and two spectra will be investigated. In this study, the coefficients of the matrix Sturm-Liouville equation will be supposed to have upper and lower triangular matrix forms and and the inverse problem will be examined by the Gelfand-Levitan method. There are two recent studies on the matrix Sturm-Liouville pencils in the finite interval which studies the inverse by the method of spectral mappings but not by the Gelfand- Levitan method. The target to be achieved in this thesis is to develop and apply the Gelfand-Levitan method which is a powerful tool to solve the inverse problem for the second order pencil of matrix Sturm-Liouville equation. If these results are reached, direct and inverse spectral problems which are still open problems, will be easily solved for some high-order Sturm-Liouville polynomial pencils.