Existence-uniqueness and reconstruction potential in the inverse Sturm-Liouville problems

In this study, firstly the Inverse Sturm-Liouville Problem which has physically great importance is considered. For the Inverse Sturm-Liouville Problems, the purpose is to reconstruct potential function by helping data which can be obtained experimentally. So generally, studies on this problems include investigation of existence-uniqueness of potential function and reconstruction potential by using data considered. In the introduction of the first chapter of this work, firstly a few examples which imply the physical importance of problem were given. Then some definitions and theorems which will be needed for next chapters were given. In the second chapter, some definitions and theorems which will be used in all study were given. Also, the work about the uniqueness of potential done by Levinson (1949) and a generalization of this work done by Hochstad (1973) were considered. In this chapter, by generalizing the notion of symmetric for symmetric potentials which can be seen in many works about uniqueness, the change of data needed depending on the degree of symmetric for uniqueness was discussed. At the third chapter, reconstruction of potential was considered. Then, the information about the reconstruction problem using zeros of eigenfunction called nodal points was given. These works contain a full set of data whereas a piece of data can be obtained in real life problems. Thus this chapter depends on the reconstruction of potential by using partial data. In this work, The problem was tried to solve by using the method of steepest descent which is an optimization method.