The Inverse Spectral Problem Of Vibration Systems

The inverse spectral problem of vibration systems is mainly concerned with the problems of determination unique and reconstruction of the vibration system when the spectral data is given, for which its application in many natural science is very important. The inverse spectral problem of vibration systems has gotten more and more attention and became one of popular research topic in computational mathematics.In this present thesis, the spectral analysis and the inverse spectral problems of three vibration systems have been studied. One kind is Sturm-Liouville prob-lem with boundary conditions involved spectral parameter, the other kind is Dirac problem with discontinuities, the third one is continuous Dirac problem.In the first chapter, we sum and review the vibration systems, especially, the backgrounds, the research significance and present situation.In the second chapter, we consider an inverse Sturm-Liouville problem with boundary conditions involved spectral parameter. The asymptotics of eigenvalues and eigenvectors are given and three sets of spectral theorem are proved.In the third chapter, an inverse problem for Dirac system with discontinuities in [0,1] is studied. It is shown that the potential function can be uniquely determined by a set of values of eigenvectors at some internal point and one spectrum(i.e. interior spectral data).In the forth chapter, the inverse problem of continuous Dirac system is consid-ered. On the basis of chapter 2, we introduce the properties of eigenvalue function, and prove three sets of spectral theorem.