Inverse Spectral Problems Of Discontinuous Sturm-Liouville Operator On Graphs And Spectral Analysis Of Laplace Operator On A Spherical Band

In 1984,O.Hald considered the half inverse Sturm-Liouville problem with one discontinuous point,indicated that one spectrum and half potential could decide the whole potential and the discontinuous information.Such study works well not only in theory,but also in many engineering applications.When detecting the structure of the earth,for example,we could decide the density of the mantle and the density jump at the crust by one torsional spectrum.When detecting the structure of a sphere,and the density function of the sphere has a discontinuous point,one spectrum could decide all the discontinuous information.Sturm-Liouville problem,on graph,is the natural extension of the classical Sturm-Liouville problem and strongly relates to many physical problems.The second chapter of this paper considers the half problem on three-star graph with one discontinuous point,gives the distribution of its eigenvalues by virtue of Rouché Theorem at first,further more,by Hadamard Factorization Theorem,indicates that one spectrum could determine part of the jump information,and finally,proves the statement that its spectrum could determine,uniquely,the whole unknow potential and all jump information when we know the potentials on two edges and half potential on another edge.The third chapter of this paper mainly considers the eigenvalue problem of Laplace operator,which enjoys significant application in quantum physics,the eigenvalues coincidence with the energy levels of the particle when particle travels through certain field,its energy can only be equal to one of the eigenvalues.The third chapter studies the Laplace operator on a spherical band with Robin boundary condition,by virtue of Maximum-Minimum Principle,indicates that when the area of the band is fixed,its first eigenvalue would be largest when the band is symmetrical about the equator,moreover,when the band's area is less than a certain number,the first eigenvalue enjoys a kind of monotonousness.