The Inverse Problem Of Two Sturm - Liouville Differential Operators

It is well known that Sturm-Liouville problem is always one of intriguing and active subjects in operator theory. Since its available, it has been used in physics, engineering and technology and other types applied subject. This article mainly dis-cusses the aftereffect Sturm-Liouville problem and the discontinuous Sturm-Liouville spectral and inverse spectral analysis problem. Specifically, this article gets the op-erator trace formula by calculating asymptotic formula of relevant meromorphic functions on the loop. Then reconstructs the kernel function and potential function by using the knowledge of spectra theory and lagrange differential formula. Lastly, applies the method of the trace formula for discontinuous Sturm-Liouville operator to discontinuous Dirac operator.The contents of this paper is arranged as follows:Chapter1the background of the problem and related boundary value problems of this paper are introduced.Chapter2the trace formulae of the Sturm-Liouville integral differential oper-ator with the Dirichlet and Neumann boundary conditions, respectively, are provid-ed. Applying the knowledge of spectra theory, the reconstruction steps of the kernel function are given.Chapter3the trace formula of the Sturm-Liouville operator with spectra pa-rameter λ in the discontinuous conditions and the reconstructing algorithm are presented.Chapter4the steps of trace formula about the Dirac operator are given.