The Inverse Spectral Problem Of Vibration Systems

Driven by the urgent need produced by the development of application in other subjects and engineering technology fields, the inverse spectral problem of vibration systems has became one of the fields that growth and develop most fast. It mainly studies, in terms of what of spectral data, the problems of determination unique and construction of the vibration system. Because of the distinct characteristics in theory, it has became a major cross topic of computational mathematics, application mathematics and system science.In this present paper, the inverse spectral problems of several vibration sys-tems have been studied, which includes Sturm-Liouville difference operator, Sturm-Liouville differential operator, Dirac differential operator and Dirac differential op-erator with jump conditions. In particular, we consider the inverse spectral problem with incomplete spectral data, the major idea lies in selecting the least spectral data to ensure the system unique.In the first chapter, we sum and review the physics backgrounds of some vibra-tion system, especially, the Sturm-Liouville system and Dirac system, and elaborate the research advances of the inverse Sturm-Liouville problems and inverse Dirac problems.In the second chapter, we consider the inverse transmission eigenvalue problem of Sturm-Liouville difference operator for absorbing media. By using the Hermite interpolation, the corresponding conditions to uniquely determined g are provided.In the third chapter, the inverse spectral problem of Sturm-Liouville differen-tial operator with boundary conditions rationally dependent on the eigen-parameter is considered. The property of spectrum of the operator is portrayed. Uniqueness results obtained that the potential function q and the parameter h in the bound-ary condition can be uniquely determined by part spectral data generated by the operator, which include part of eigenvalues and part of norming constants on the condition that the boundary condition function f(λ) is Herglotz function and which is known prior. In the forth chapter, the inverse spectral problem based on the blend spectral data for the Dirac differential operators is considered. Some uniqueness results are obtained which imply that the pair of potentials (p(x), r(x)) and a boundary condi-tion are uniquely determined even if only partial information is given on (p(x), r(x)) together with partial information on the spectral data. Moreover, we also concern with the problem of missing eigenvalues when p(x) and r(x) are Cn-smoothness at critical point, that is, by using the n order derivative exchange n eigenvalues in order to ensure the system unique.In the fifth chapter, the three spectra inverse problem of the Dirac differential operator is considered, by using the monotonicity of the Weyl-function on real axis, the alternation of the three spectra is obtained, we prove that, the three spectra uniquely determine the pair of potentials (p(x),r(x)) and the parameters h, H in the boundary conditions if the two spectra of the operators defined on subsets are disjoint.In the sixth chapter, the inverse spectral problem for non-selfadjoint Dirac operator with eigenvalue dependent boundary and jump conditions is considered. The uniqueness theorems are proved, and the conclusion state that Weyl-function, and generalized spectral data, are equivalence in proving the uniqueness of potential function; The constructive procedure for solving the inverse problem are given, which include from Weyl function, and generalized spectral data, respectively.