Semi-inverse Problem Of Discontinuous Sturm-Liouville And Dirac Operators

With the development of mathematical physics,electronics,geophysics and other natural sciences,the related problems of discontinuous Sturm-Liouville operator and Dirac operator have gradually become an attractive research field in Applied Mathematics.In this thesis,we study the uniqueness,certainty and reconstruction algorithm of a class of Sturm-Liouville operator with finite impulse points of the potential and a class of Dirac operator with discontinuous eigenfunction when different mixed spectral data are known.The main research contents of this thesis are as follows:In the first chapter,the research backgrounds and meanings of Sturm-Liouville operator and Dirac operator,the research status of their half-inverse problems and the major target of this thesis are introduced.In the second chapter,the half-inverse problem of Sturm-Liouville operator with finite impulse points of the potential and Robin boundary conditions is studied.Given all the eigenvalues but one,together with the potential on the half-interval and one boundary condition,the method of Pivovarchik is used to discuss the existence and uniqueness of the potential on the other half-interval and the other boundary condition.The corresponding reconstruction algorithm of the unique solution is given.In the last chapter,the half-inverse problem of Dirac operator with the jump conditions is studied.A method of recovering the potential on a half-interval using a known potential on another half-interval and the spectrum on the whole interval is proposed.