According to the two spectra identification of sturm-liouville operators

This thesis consists of six chapters. In the first chapter, historical development and literature review of subjects are examined. In the second chapter; some fundamental definitions and theorems often used in theory of spectral of differential operators are given. In the third chapter; classic Sturm-Liouville equation , this equation of serial expansion respect to eigenfunctions are gived , and classical Fourier Integral is studied. In the fourth chapter; Fourier-Bessel serial expansions, Fourier-Hankel Integral expansions are given. In the fifth chapter; Legendre Polynomials are defined, the ortogonality of them are showed, eigenvalue and normalization eigenfunction of them are found. In the six chapter; Trace formulas are obtained for Sturm-Liouville operators and for linear damped wave equatıon. Key words: Sturm-Liouville problem, eigenvalue, eigenfunction, Legendre functions, Bessel functions, trace formulas.