Self-adjointness Of The Product Of Hamiltonian Operators Under The Limit Circle Case

The research of the self-adjointness of differential operator and the self-adjointness of product of differential operators has been obtained some good results. Since each standard self-adjoint differential operator can be written as a Hamiltonian system. As a result, we can study the self-adjointness of Hamil-tonian operator in light of the related theory and methods of the self-adjoint extention for differential operator.In this paper, the research method is different from the that of [30]. This paper is concerned with self-adjointness of the product of Hamiltonian opera-tors under the limit circle case. The method used in this paper is different from that in paper [30]. Using the Calkin method and the construction of self-adjoint extensions for singular Hamiltonian systems, the necessary and sufficient condi-tions which make the product of Hamiltonian operators under the limit circle case being the adjoint operators are obtained.According to the contents, the text is divided into three chapters:Chapter 1 Introduction and preliminaries, in this chapter, we consider the following linear Hamilton system: Jy'(t)=(λW(t)+H(t))y(t), t (?) [a,b), (1.2.1) (where a is a regular point, while b is singular, i.e b=+∞or W(t) or H(t) is not integrable near b, W(t) and H(t) are 2n x 2nHermitian matrices and integrable on [a, b), J is the canonical symplectic matrix, i.e , and In is the n x n unit matrix, W(t)> 0 is a weight function), and L is the Hamilton operator corresponding to system (1.2.1), i.e: What is more, we briefly introduced the relevent theory study background and re-sults of the Hamilton operator, gave the related conclusions of self-adjoint Hamil-ton operator. Chapter 2 In this chapter, the content is divided into two parts. At first,we give the preliminaries of the self-adjointness of the product for two Hamiltonian operators under the limit circle; Then, the main result based on the first part is obtained.Chapter 3 Based on the above consideration, we further study the self-adjointness for the product of the three Hamiltonian operators, and we give out the main result that the necessary and sufficient conditions which make the product of three Hamiltonian operators under the limit circle case being the adjoint operators are obtained.