Study On Inverse Problem Of Discontinuous Sturm - Liouville

It is well known that spectral theory of Sturm-Liouville problems have been used in physics, engineering technology and other types applied subjects. Many famous mathematicians and physicists have studied it in depth and broad. This thesis studies the corresponding inverse of the discontinuous Sturm-Liouville problems. For this problem, this thesis gives not only the reconstruction formula of potential function, but also for two cases of the trace formula in the boundary conditions with and without spectral parameters.The contents of this paper is arranged as following:Chapter1Introduce the background of the problems and main contents of this paper.Chapter2Study the problem of the reconstruction of potential. By virtue of and by using a spectrum and potential function known on the half interval and Lagrange differential formula, reconstructing formula of potential functions on the whole interval is given.Chapter3Study the trace formula of the discontinuous Sturm-Liouville problems. Taking use of the information of spectral theory, the property of holomor-phic function, translation operator and the skill of appropriate integral road, the trace formulae in the boundary conditions with and without spectral parameters are given.