The Inverse Spectral Problem Of Two Vibration Systems

The inverse spectral problem of vibration systems mainly be concerned with the uniqueness and reconstruction of the vibration systems under the condition of that spectral data is given. The application of this problem is very important in many natural sciences. Therefore, the inverse spectral problem of vibration systems has gotten more and more attention and became one of popular research topics in applied mathematics.In this present thesis, the spectral analysis and the inverse spectral problems of Sturm-Liouville system and Dirac system have been studied. The main contents are as following:In the first chapter, we sum and review the research backgrounds, significance and advances of the inverse Sturm-Liouville problems and inverse Dirac problems.In the second chapter, we provide the eigenvalue asymptotics for Sturm-Liouville problem with the interior point and boundary conditions are involved spectral pa-rameter and Dirac problem with piecewise smooth potential function, respectively.In the third chapter, the inverse spectral problem of Sturm-Liouville problem with the interior point and boundary conditions are involved spectral parameter is considered. We prove that the potential function is uniquely determined by Weyl function or two groups of spectrum or generalized spectral data, and provided the reconstruction method for this system by virtue of given spectral data.In the forth chapter, the inverse spectral problem of Dirac problem with piece-wise smooth potential function is considered. We introduce the properties of eigen-value function, prove that the potential function is uniquely determined by Weyl function or two groups of spectrum or spectral data, and provided the reconstruc-tion method for this system by virtue of given spectral data.