Research On Two Classes Of Differential Operators And The Construction Of Riesz Basis

The differential operator is a kind of unbounded linear operators,which has widely research fields,including deficient index theory,self-adjoint extentions,numerical meth-ods,completeness of eigenfunctions and the dependence of eigenvalues,coercivity,asymp-totic estimation and inverse spectral problems.This thesis focuses on the self-adjointness and the dependence of eigenvalues on the problem of third-order differential operators,the multi-point discontinuous Sturm-Liouville problem,the construction and stability of Riesz basis.Firstly,we investigated a class of third-order differential operators with eigenparam-eter dependent boundary conditions,where the boundary conditions are in the sense of mixed,that is,two boundary conditions are separated and one boundary condition is coupled.Firstly,using operator-theoretical formula,we construct a new inner product,and in the frame of new Hilbert space,we transfer the boundary value problem to a operator form.The self-adjointness,the properties of eigenvalues and eigenfunctions of the operator are investigated.An entire function is constructed such that its zeros are the eigenvalues of the operator.Its Green function is also calculated.In the end,we prove the dependence of the eigenvalues of the operator on the partial coefficients of the equation and the coefficient matrix of the boundary conditions,and obtain the differential expressions of the eigenvalues with respect to the given coefficients and matrices.Secondly,we study a class of multi-point Sturm-Liouville problems with finite dis-continuous points,and with abstract functionals in the equation and boundary conditions.It should be noted that the boundary transmission conditions are in the sense of coupled since the usual is in the sense of separated.For this kind of problems,using the theory of ordinary differential equation,we give the existence and uniqueness of solutions.Then we establish the isomorphism,Fredholmness and coerciveness with respect to the spectra parameter by studying the principle part of boundary value problems and show that the resolvent of the operator corresponding to an irregular problem decreases with respect to the spectral parameter,but there is no maximal decreasing at infinity.Finally,two sequences related to sine function and cosine function are constructed,it is proved that these two sequences form Riesz bases in L~2[0,?] by using the boundedness and completeness of the sequences,moreover,the stability is discussed.As an applica-tion,we study Riesz basis problem for Sturm-Liouville problems with Dirichlet boundary conditions,it is proved that a sequence consisting of eigenfunctions form a Riesz basis in L~2[0,?].The whole thesis consists of five parts:1.The research background and main results of this paper;2.Self-adjoint realization and Green's function for a class of third-order differential operators with eigenparameter dependent boundary conditions;3.The depen-dence of eigenvalues for a class of third-order differential operators with eigenparameter dependent boundary conditions;4.Solvability and coerciveness of multi-point discon-tinuous Sturm-Liouville problems with abstract linear functionals;5.Construction and stability of Riesz bases.