Spectral theory of the singular Sturm-Liouville operator

II ABSTRACT PhD Thesis SPECTRAL THEORY OF THE SINGULAR STURM-LIOUVILLE OPERATOR Hikmet KOYUNB AKAN Fırat University Graduate School of Science and Technology Department of Mathematics 2002, Page: 84 This thesis is arranged in five chapters. In the first chapter, basic definitions and theorems that are given. In the second chapter, Legendre Equations, Legendre polinomals and conjugate Legendre functions and informations of the relevant is given. Asiymptotic formulas for the eigenvalues, eigenfunctions, normalized number of the Sturm-Liouville operator that has singularity both ends interval are found. Furthermore, for these equations Oscillation theorem is proved. In the third chapter, for the singular differential operator having different potentials, existence of the translation operator is given. Hence, second order singular Hyperbolic equation which obtained for nucleus K(x,t) is transformed to Euler- Poisson-Darboux equation and it is solved by Riemann method. In the fourth chapter, on the point similiar to Marchenko theorem, uniqueness spectral function of the singular Sturm-Liouville operator is proved. Later, Levitan- Marchenko integral equation that known basic integral equation of the inverse problem is obtained. In the fifth chapter, Inverse problem with peculiarity in two partially non coincided spectrum is examined. Using the proof of the Hochstadt method for singular differential operator, concerning the structure of the difference q(x) - q(x) is given. Finally, When the determined conditions is supplied for definiting potantial q{x) according to the a single spectrum sufficent theorem is proved. Key words: Eigenvalue, Eigenfunction, Spectrum, Riemann Method, Basic Integral Equation.