The Self-adjointness And Eigenvalue Of A Class Of 4-order Differential Operators With Transmission Conditions

In this paper , the self-adjointness, eigenvalue problem and completeness of eigenfunctions of the differential operator with a discontinuous point in the interval, namely, a class of fourth-order differential operator with transmission conditions were study. By constructing the inner product associated with the transmission conditions, studying this issue in the Hilbert space associated with the transmission conditions, and defining the maximum and minimum operator associated with transmission conditions, proving that they are conjugate each other, and giving that such differential operators for the self-adjoint operators of the necessary and sufficient conditions. At the same time constructed an entire function, and proved that the zero point of this entire function of eigenvalue of this problem, such problem have been at most a countable eigenvalues, and there is not finite value of the accumulation point. Also according to the connection which transmission conditions established on the solutions of the two intervals, given the Green function, and combined with the spectral theory of compact operators, as well as the corresponding nature of inverse operator, proved that the eigenfunction systems is complete. This paper is divided into four chapters. The first chapter: introduction. The second chapter: the characterization of the self-adjointness of a class of 4-order differential operators with transmission conditions. The third chapter: the eigenvalue problem of a class of 4-order differential operators with transmission conditions. The fourth chapter: completeness of eigenfunctions of a class of 4-order differential operators with transmission conditions.