The Inverse Spectral Problem Of Two Types Of Vibration Systems

The inverse spectral problem of vibration systems is mainly concerned with the uniqueness and reconstruction of the vibration systems under the condition that spectral datas are given.The application of this problem is not only important in mathematics but also has a wide application in mechanics,electrics,spectroscopy,natural science and engineering problems.Therefore,the inverse spectral problem of this vibration systems has gotten the great attention of experts and scholars and became one of the most popular research in mathematics.In this paper we will study the inverse spectral problem of continuous and discrete vibration systems,where the discrete and continuous vibration systems refer to Jacobi matrices associated with mass-spring systems and regular Sturm-Liouville system related to string equations,respectively.The main works are given as follows:In the first chapter,we summarize the research backgrounds,significance and advance of the inverse spectral problem of regular Sturm-Liouville system and Jacobi matrices and introduce the relationship between them.In the second chapter,inverse spectral problem of Jacobi matrices is considered with a missing eigenvalue.The necessary and sufficient conditions for their solvabil-ity of the problem are derived.Furthermore,we introduce the application of the theorem in mass-spring system and show the mumerical algorithms of constructing this system.In the third chapter,inverse spectral theorem of regular Sturm-Liouville system is considered.Some average formulae for the nodal domains are derived,and the algorithm of constructing the density function of regular Sturm-Liouville system is provided.