Properties For Two Kinds Of Discontinuous Diferential Operators With Eigenparameter Dependent Boundary Conditions

As is known to all,the spectral theory of differential operator plays an important role in mathematical theory.It permeates into all fields of mathematics,among which one of the most widely used operator is Sturm-liouville operator.With the extensive study of the spectrum theory of ordinary differential operators,the contents of the spectrum of operators have greatly developed,especially on the eigenvalues of the spectrum has made great progress.More and more researchers have paid close attention to it and its inverse problem,eigenvalue and asymptotic formulas for the eigenfunctions,completeness of eigenfunctions and Green function.In the second chapter of this article,we consider inverse eigenvalue problems for a discontinuous Sturm-Liouville operator with finite discontinuities.The inverse problem of discontinuous Sturm-liouville particles with spectral parameters on the boundary condition has been studied by many authors,the background and status of Sturm-liouville equation are introduced at the beginning of this paper.We define the operator in a suitable Hilbert space,such that the eigenvalue of the boundary value problem is consistent with this problem.We extend and generalize many approaches and results of the classical regular Sturm-liouville problems to similar problems with discontinuities.We construct its fundamental solutions,discuss some properties of its spectrum and get the asymptotic formulae.We define the Weyl function finally,and prove the uniqueness theorems of spectral problems.In the third chapter of this article,we research a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions.A new operator associated with the problem is established,so that the eigenvalue of the boundary value problem is consistent with this problem,the self-adjointness of the operator is proved.We give asymptotic formulas for the basic solutions and the characteristic function and research the completeness of eigenfunction and then finally we have Green function.