The Inverse Three Spectral Problem For Discrete And Continuous Vibration Systems

The inverse spectral problem of vibration systems mainly be concerned with the uniqueness and reconstruction of the vibration systems under the condition that spectral data is given. The application of this problem not only is important in mathematics but also have a wide and direct application in physics and natural science. Therefore, the inverse spectral problem of this vibration systems has gotten more and more attention and study by mathematician and physical and became one of the popular research in applied mathematics.In this paper we will talk about the inverse spectral problem of continuous and discrete vibration system, where the discrete and continuous vibration systems refer to Jacobi matrices and regular Sturm-Liouville system, respectively. The main works are given as follows:In the first chapter, we sum the research backgrounds, significance and advance of the inverse three spectral problem of regular Sturm-Liouville system and Jacobi matrices and briefly introduce the relationship between them.In the second chapter, inverse three spectral theorem of Jacobi matrices is considered. we prove a new inverse three spectral theorem for Jacobi matrices with a rank 2 perturbation of the matrix T. At the same time we introduce the application of the theorem in mass-spring system which turn out to be unique and can be constructed.In the third chapter, inverse spectral theorem of regular Sturm-Liouville system is considered. we provide a new way to prove the uniqueness and construct the potential of regular Sturm-Liouville system when the three spectra on intervals [0, ?] and [?,1] are overlapped.