Research On The Dissipation, The Dependence Of Eigenvalues On The Problem For Some Classes Of Differential Operators With Interior Discontinuity

In recent years,along with numerous questions arising in the application fields,more and more researchers focus on the problem of the solutions and their derivative with discontinuity inside an interval,and eigenparameter dependent boundary value problems of differential equations.These kinds of problems root in many physical,even medical problems,for example,the problems of string vibration with a node and diffraction of light,etc.Many actual physical problems can be transferred to the study of interior discontinuous differential operators.To deal with the discontinuity of these problems,general method is adding transmission conditions on the discontinuous points to realize the relation of solutions at such points between adjacent intervals.In this paper,we focus on the spectral analysis of several types of differential operators with discontinuity,and the research contents are divided into two aspects: in the first aspect,the dissipation properties,discreteness of spectrum and eigenfunction expansion for several kinds of differential operators with discontinuity are studied;the second aspect focuses on the dependence of eigenvalues of two classes of higher order self-adjoint differential operators with discontinuity on the problem.In the early 1969 s,I.C.Gohberg and M.G.Krein in their famous book “ Introduction to the Theory of Linear Non-self-adjoint Operators ” [49] investigated some famous theorems such as Krein theorem,Liv?sic theorem,etc,which are used to the study of the problem of eigenfunction expansion for the dissipative operators.In 1970 s,B.S.Pavlov [86] proposed a new method for the spectral analysis of dissipative operators.This method is based on building their self-adjoint dilation and the corresponding functional model of Sz.-Nagy-Foia?s type.Then one could study spectral properties of the operator using the equivalence between the Lax-Phillips scattering matrix and the characteristic function.Using the aforementioned methods,we study the dissipation for a class of singular Surm-Liouville operators with discontinuity inside an interval.In the case of one endpoint is regular and the other endpoint is singular limit circle,uniting the boundary and transmission conditions,by constructing a new Hilbert space,and transferring the considered problem to operator form in the new space,we prove the operator is a dissipative operator and obtain some properties of its eigenvalues.By constructing the solutions which satisfy the equation and boundary transmission conditions,the characterization of the eigenvalues as zeros of an entire function is established.And obtain its inverse operator by calculating the Green function.Using Liv?sic theorem,we prove the system of eigenfunctions and associated functions is complete in the constructed Hilbert space.Using Pavlov's method,we study the dissipation and eigenfunction expansion of singular Dirac operators with eigenparameter dependent boundary condition.Since eigenparameter appears in the boundary condition,general Hilbert space does not appropriate to the study of the problem.Using operator theory,we introduce a new inner product associated with the eigenparameter in the boundary condition,and pass the considered problem to a maximal dissipative operator ?in the new Hilbert space.The self-adjoint dilation ?of ?in the space ? is constructed.In light of Lax-Phillips scattering theory,the incoming and outgoing spectral representations of ?are constructed such that the scattering matrix is obtained,the functional model of the maximal dissipative operator is constructed,in view of the scattering matrix of dilation,we then derive the characteristic function.Using the equivalence between scattering matrix and characteristic function,and the theory of characteristic function,the discreteness of spectrum and completeness theorem on the system of eigenfunctions and associated functions of this dissipative operator are proved.In order to further study the problem,we then consider the dissipation and eigenfunction expansion of discontinuous singular Dirac operators with eigenparameter dependent boundary condition.Since the discontinuity of the problem,using transmission conditions imposed on the discontinuous point,we construct a new Hilbert space associated with the problem.In the framework of new space,we need to reconstruct associated operator and the self-adjoint dilation of the maximal dissipative operator,these are all closely related with the coefficients of transmission conditions.Through modified Pavlov's method,we prove the discreteness of spectrum and completeness of the system consisting of eigenfunctions and associated functions.We also study the eigenvalue problems of a class of discontinuous fourth order beam equation.Under some classes of special boundary conditions,we prove that the eigenvalues dependent not only continuously but also smoothly on the parameter of the problem.In particular,we obtain a differential expression of an eigenvalue with respect to a given parameter.The dependence of eigenvalues and eigenfunctions with respect to the data plays an important role in the theory of differential operators,it gives theoretical support for the numerical computation of eigenvalues.In order to generalize the above results to a more general situation,we study the dependence of eigenvalues of 29)th order real symmetric differential operators with discontinuity on the problem.We prove that if is an eigenvalue of the considered problem,then can be embedded in a continuous eigenvalue branch,and is a differentiable function with respect to a given parameter.Moreover,we give a differential expression of an eigenvalue with respect to a given parameter,the properties of monotonicity of eigenvalues with respect to the parameter can be obtained via the derivative of eigenvalue on a given parameter.