The Self-Adjointness,Dissipation And Spectrum Analysis Of Some Classes High Order Differential Operators With Discontinuity

In recent years, more and more mathematical and physical researchers pay attention to a class of differential operators with discontinuity in the interior point, and the differential operator problem for the differential equation and boundary conditions with eigenparameter. A lot of actual physical problem can be transformed for the interior discontinuity differ-ential operator problems, in the fields of engineering and technology, some partial differential equations by the method of separation of variables can be transformed into the boundary conditions with eigenparameter of the differential operator problem, and some problem need to be converted to high order case to handle, so it is very important to investigate the self-adjointness and spectrum of the high order differential operators with transmission conditions and eigenparameter boundary conditions. Dissi-pative operator is a class of very important nonself-adjoint operator in operator theory, and also has a strong application background. In this paper, we investigate the self-adjointness and spectrum analysis for the fourth-order and high order differential operators with discontinuity in the interior point and with eigenparameter-dependent boundary conditions, and the completeness of eigenfunctions and associated functions for dis-continuous fourth-order dissipative operator, the interior discontinuity is characterized by the transmission conditions of the problem. First, we investigate a class of high order differential operator with discontinuity at an interior point in the interval, it contains two parts. In the first part, we investigate the self-adjointness of2nth order differential operator with transmission conditions, while the coefficients of boundary conditions and transmission conditions are determined by matrix, and sat-isfy certain conditions, by the definition of self-adjoint differential operator, and by matrix representation to simplify the problem, we prove the opera-tor is self-adjoint, and all eigenvalues are real, to the different eigenvalues, the corresponding eigenfunctions are orthogonal. In the second part, we investigate the necessary and sufficient conditions of the self-adjointness for2nth order differential operator with transmission conditions, give the gen-eral boundary conditions and transmission conditions, and the coefficient matrix are complex. We deal with them in a Hilbert space associated with the transmission conditions, by the general theory of differential operator, we obtain the necessary and sufficient conditions of self-adjointness, and require the values of the determinants are equal to(not equal to zero).Second, we investigate a class of discontinuous fourth order differ-ential operator with eigenparameter-dependent boundary conditions prob-lem which has a wide range of applications in the fields of engineering and technology. While two boundary conditions with eigenparameter, in a suitable Hilbert space, we define a linear operator associated with eigen-parameter, such that the eigenvalue of this problem is equal to the discon-tinuous fourth-order differential operator with eigenparameter-dependent problems, i.e. we transform this problem to study the eigenvalues and eigenfunctions in this new Hilbert space, we prove the operator is self- adjoint, all eigenvalues are real, to the different eigenvalues, the corre-sponding eigenfunctions are orthogonal in the sense of corresponding inner product; to the discontinuous fourth order differential operator with four eigenparameter-dependent boundary conditions, we give the general trans-mission conditions and eigenparameter-dependent boundary conditions, by the coefficients of boundary conditions and transmission conditions we con-struct two fourth order matrix, by the general theory of self-adjoint oper-ator, we obtain the necessary and sufficient conditions of self-adjointness, then we construct basic solutions, and obtain the entire function which concern the eigenvalues, and have the conclusion:the complex number λ is an eigenvalue of the operator if and only if the entire function is equal to zero; then, we prove the operator has only point spectrum, and obtain the Green’s function of the operator, and it is different from the case of general boundary conditions.Then, we investigate a class of2nth-order differential operator with transmission conditions, boundary conditions and n eigenparameter-dependent boundary conditions, in the instances of coefficients of eigenparameter-dependent boundary conditions, we ingeniously construct n second-order determinants, and define inner product by the values of these second-order determinants, and define the operator associated with eigenparame-ter, transform the problem into investigate the eigenvalues of operator A, while the boundary conditions and transmission conditions satisfy some conditions, we prove the self-adjointness of the operator, obtain the entire function which concern the eigenvalues and we have the conclusion:the eigenvalues of the problem coincide with the zeros of the entire function detΦ(1,λ), and last we obtain the operator has only point spectrum.In the last part of this paper, we investigate a class of discontinu-ous fourth-order dissipative operator, we give the boundary conditions and transmission conditions, where the regular point a satisfy general boundary conditions, the coefficients of boundary conditions for the singular point b satisfy certain conditions, by definition of dissipative operator, we prove this operator are dissipative in the Hilbert space with special inner prod-uct, and have no real eigenvalues; then we obtain the entire function Δ(λ) which concerned the eigenvalue; to obtain the inverse of operator A, we determine the Green’s function by calculation, and prove zero is not the eigenvalue of A; in the last, by the Livsic theorem, we prove the eigenfunc-tions and associated functions of operator A are complete in the space H, and has infinitely many eigenvalues.This paper contains seven parts. The first part:an introduction of the background we investigate and main results we obtain in this paper. The second part:the self-adjointness of high order differential operator with transmission conditions. The third part:the necessary and sufficient conditions of self-adjoint high order differential operator with transmission conditions. The fourth part:the self-adjointness of fourth order differential operator with transmission conditions and two eigenparameter-dependent boundary conditions. The fifth part:the the necessary and sufficient con-ditions and completeness of eigenfunction for fourth order differential op-erator with transmission conditions and four eigenparameter-dependent boundary conditions. The sixth part:the self-adjointness and complete-ness of eigenfunctions for2nth-order differential operator with transmission conditions and eigenparameter-dependent boundary conditions. The last part:the discontinuous fourth order dissipative operator and the complete-ness of the eigenfunctions and associated functions.