Direct and inverse problems of spectral analysis for a class of differential operators

In this study, direct and inverse problems of spectral analysis of two different boundary value problems defined on the positive semi axis have been investigated. In the first part of the thesis, the inverse problem of the scattering theory for the boundary-value problem involving the spectral parameter in the boundary condition is examined. For this problem, i) Scattering data has been defined. ii) The characteristics of the scattering data have been examined. iii) The uniquely construction algorithm of the potantial with scattering data has been given. iv) The resolvent operator has been constructed. v) Expansion formula with respect to eigenfunctions has been obtained with the method of Titchmarsh. vi) Gelfand-Levitan-Marchenko main equation has been obtained for the solution of the inverse problem. In the second part of the thesis, the boundary value problem for differential equation with discontinuous coefficients has been considered. For this problem, i) Special solutions have been defined and the properties of the solution have been examined. ii) Scattering data has been defined and the characteristics of the scattering data have been examined. iii) Resolvent operator has been constructed. iv) Expansion formula for singular Sturm-Liouville operator has been obtained. v) The main equation has been obtained.