The stability problems in spectral theory of di¤erential operators

This thesis will provide important contributions to the theory of the stability of differential operators. For the problems considered in this thesis, when the eigenvalues coincide the finite numbers, it is aimed to obtain the finite bounds for the difference between the spectral functions and eigenfunctions. The method which used was applied firstly by T.I. Ryabushko [60] to demonstrate the stability of regular Sturm-Liouville problems. This thesis consists of seven chapters. In the first chapter, the history of the spectral theory of regular and singular Sturm-Liouville operator, Dirac and Diffusion operators is given. In the second chapter, some basic definations and theorems which are frequently used in spectral theory of differential operators are given. In the third chapter, important applications of the transformation operator for the differential operators are given. This method was applied in the fourth, fifth and sixth sections which are the original parts. Following results were obtained respectively. In the fourth chapter the problem of stability for Dirac operators is defined and the formula explaining the norming constants respect to two spectra is given. The stability theorems have been proved. In the fifth chapter the problem for Sturm-Liouville operators with Bessel type singularity with Dirichlet boundary conditions have been considered. The stability theorem according to two spectra is given for this problem. In the sixth chapter, Diffusion equation with discrete boundary conditions is considered. The formula explaining the norming constants of this problem respect to two spectra is given. The stability theorems have been proved. In the seventh chapter the results of the thesis are evaluated.