J-self-adjoint Differential Operator And Its Spectrum Analysis

Recently, the theory of non-self-adjoint differential operators have been concerned by people, and many mathematicians have turned their studies to it, because it is widely used in the dissipative problems of energy, the theory of inverse scattering, the inverse spectra and other interesting problems.In the 50's of last century , Glazman noticed a class of special non-self-adjoint operator, the J-self-adjoint operator. He investigated the spectral theory of J-self-adjoint operators, and obtained some important results about J-self-adjoint operators.In 1957, Sims studied second order linear differential equations with a complex coefficient by the method of Weyl , and obtained an analogue for it with a real coefficient of the limit-point, limit-circle theory.The main objective of this paper is to extend and complete the Sims limit-point, limit-circle classification on second order linear differential equations with a complex coefficient , and to study its Weyl function and Weyl solution , then we obtain some new results.In this paper, we also obtain a sufficient condition for the discreteness of spectrum of 2n-th order J-self-adjoint differential operators.This paper contains four parts. The first part: an introduction of the background and advance of the study on the J-self-adjoint differential operators.The second part : the limit-point, limit-circle theory of second order J-symmetric differential operators.The third part: we investigate the Weyl function and Weyl solution on second order J-self-adjoint differential operators.The fourth part: we obtain a sufficient condition for the discreteness of spectrum of 2n-th order J-self-adjoint differential operators.