The Resolvent Operator And Spectrum Analysis Of Vector Differential Operator

One of the basic problems of the theory of differential operators is the spectrum theory. The studying methods are various.One of them is by the studying of the Green function of the resolvent operator to study the spectrum,characteristic determinant,eigenfunctions and their expansion.The spectrum theory of differential operators is specially concerned because it has a close relationship with application.Especially after the appearance of the criterion for the discrete spectrum of two-order self-adjoint differential operator,many results about spectrum were published.But there is few studying results about the vector differential operators.The resolvent operator of vector differential operator, the properties of its kernel Green function and the criterion for the discrete spectrum are studied in this paper.This paper is divided into four parts.In the first part,the background and development of differential operator (vector differential operator) theory are introduced simply.In the second part,We study the resolvent operator of 2n-order J-selfadjoint vector differential operator which is regular at the ends a and b.The resolvent operator is an integral operator.Then, we obtain some analytic properties of the kernel of the resolvent operator (Green function).Finally, we give an estimation for Green function and obtain the results that the resolvent operator is complete continuity operator and its spectrum is discrete.In the third part,by the method of analysis,we study the resolvent operator of 2n-order J-selfadjoint vector differential operator with one endpoint singularity and obtain some properties of the resolvent operator.Two sufficient conditions for the discreteness of two-order self-adjoint vector differential operator are given in last part.