Inverse Spectral Analysis Of The Sturm-Liouville Problem And The Dirac System

The differential operator mostly researches these two problems, one is to research the spectrum problem, and the other is to research the inverse spectrum problem. The inverse spectrum problem is by the information of the spectrums especially the eigenvalues can determine the differential operator and reconstruct it. In this paper, the inverse spectral problems of the Sturm-liouville operator and the Dirac operator will be studied. The work of this paper is as follows:Firstly, as the preparation of the inverse problem, the basic knowledge of the Sturm-liouville operator and the Dirac operator will be given. It includes the investigative history, the eigenvalues and the eigenfunctions of the two operators. And then, the definition and the proposition of the Weyl function will be given based on the two operators.Secondly, we determine the Sturm-liouville operator. For the Sturm-liouville expression, based on the Dirichlet boundary conditions and non-Dirichlet boundary conditions, given a certain amount of eigenvalues, we can uniquely determine the potential functions applying the Weyl function method under the potential functions. At the same time, the Sturm-liouville system will be determined. Besides, the conditions of the potential functions decide the numbers of the eigenvalues.Thirdly, we determine the Dirac system. For the Dirac operators, given a certain amount of eigenvalues, we can uniquely determine the potential functions applying the Weyl function method under the corresponding boundary conditions and the potential functions. At the same time, the Dirac system will be determined. Besides, the conditions of the potential functions decide the numbers of the eigenvalues. We will study the problems in the case of the information known on few or a majority interval of the potential functions respectively.