The Research Of Some Operators In A Kein Space

The theory of differential operators which has profound theoretical significance and practical significance is the theoretical basis for mathematical physics, quantum mechanics and other engineering disciplines. In order to solve the problems in the differential equations, differential operators theory provides a theoretical framework and a solution. This paper will study two types of differential operators. The one class is a second order differential operator with transferring conditions which are defined in a Krein space, the other class is the left-definite four-order differential operators.In the introduction section, we will briefly introduce the research status, the research purpose and research significance of differential operator theory.Chapter I, In order to understand this article better, this chapter will introduce some basic concepts and theories, such as the theory of differential operator, Krein space, left-definite differential operators and right-definite differential operators.Chapter II, In the section, the paper mainly studies a sort of Sturm-Liouville operators with transferring conditions which are defined in a complete Krein space. Finally, when we study the problem in a Krein space, the operator is proved to be self-adjointness.Chapter III, In the section, we will study an algorithm to compute eigenvalues of the problems of left-definite four-order differential operators. Our primary task is translating left-definite four-order differential operators into right-definite four-order differential operators of dimension four and then giving an algorithm to compute eigenvalues of the problems of left-definite four-order differential operators.Chapter IV, In the section, we will summarizes the main achievements of this article. The content of the article is a small part of the content of the differential operator. At last we list some problems remaining to be solved.