Spectral properties of some differential operators on time scales

In this thesis, we try to get some basic features of di¤erent types of operators on time scale. This thesis includes six chapters. In chapter 1, historical improvement of time scale theory is given. In chapter 2, some fundamental properties, some notions and theorems of time scale calculus are explained. Then, first and second canonic forms of Dirac system in classical spectral theory are expressed on time scales in chapter 3. The physical meaning of Dirac system is stated. Therefore, some important properties of Dirac system are given on a time scale for the first and second canonic forms are given. Furthermore, the asymptotics estimate of the eigenfunction for the first and second canonic forms of Dirac system are found. In chapter 4, impulsive diffusion equation on time scales is studied. Before expressing the main results, the physical meaning of impulsive di¤usion equation and some properties of this equation in spectral theory are given. Then, the basic properties of impulsive diffusion equation are generalized to an arbitrary time scale. And, asymptotic estimate of the eigenfunction for impulsive di¤usion equation on a time scale is constructed. Eventually, Bessel equation on time scale is considered in chapter 5. Its physical properties are defned. Then, its basic properties are expressed on time scale. Moreover, the asymptotic estimate of the eigenfunction for Bessel equation is given. In chapter 6, some fundamental conclusions about thesis are given.