On Products Of Self-adjointness Of Some Classes Of Differential Operators

Ordinary differential operators theory can supply differential equations, classical physics, modern physics and other technique fields the theory basis, which is a compositive and edging mathematics banch of ordinary differential equations, functional analysis, space theory, operators theory etc.. I t contains a great important problems, such as deficiency index theory, adjoint extension, spectral analysis, press eigenfunction to spread, numerical method, inverse questions, and so on.On self-adjointness of differential operators product have obtained some results ([22],[23],[24]). In this paper, we study an important problem in the field of differential operators of self-adjointness and some discussions as follows: making use of general structure theory of differential operators, we disscuss the self-adjointness of the product of a class of high-order limit-point differential operators and a class of high-order limit-circle differential operators, and obtain the sufficient and necessary conditions that boundary conditions satisfy and some other results; and meanwhile, obtain the sufficient and necessary conditions of the self-adjointness of the product for three high-order limit-point differential operators and three second-order limit-circle differential operators, and some corresponding results. This paper contains four chapters.Chapter one is divided into two parts: in the first part, we give the simple summarize of differential opertors theory; in the second part, we give the basic knowledge of symmetric differential operators.Chapter two is divided into two parts: in the first part, we discuss the self-adjointness of the product of two mth-order limit-point differential operators; in the second part, we discuss the self-adjointness of the product of three high-order limit-point differential operators, and obtain the sufficient and necessary conditions which can make the product operators being adjoint and some other results.In Chapter three, we discuss the self-adjointness of the product of three second-order limit-circle differential operators, which contains regular and singular two cases, and the sufficient and necessary conditions and some other results are obtained.In Chapter four, we discuss the self-adjointness of the product of a class of high-order limit-circle differential operators, which contains regular and singular two cases, the sufficient and necessary conditions which can make the product operators being the adjoint operators are obtained.