A New Description Of Self-adjoint Extensions Of Linear Hamiltonian Operators

Hamilton principle is appeared in the form of Hamiltonian systems in the fields of mathematical science, life science and so on, especially many of the mathematical models of quantum mechanics and biological engineering. Hence, researching on theory of linear Hamiltonian systems is of theoretical and practical significance.Identification of the self-adjoint extensions and characterizations of all the self-adjoint extensions of minimal Hamiltonian operators have attracted a great deal of interest of many scholars because spectral theory of self-adjoint differential operators has become one of the hot topics of mathematics. Professor Zhijiang Cao gave a com-plete and direct characterization of all the self-adjoint extensions in terms of solutions for the second-order and high-order symmetric differential operators in the limit circle case (see [1,2]). Professor Jiong Sun gave a complete and direct characterization of all the self-adjoint extensions for high-order symmetric differential operators with middle defect indices (see [3]). Professor Guangsheng Wei give a new description of self-adjoint domain of symmetric operators (see [4]). If the deficiency indices of closed symmetric operator To is (m,m), then Im(T0*y,y) is a quadratic form of rank 2m. Using this characterization, we get a new method of self-adjoint extension of To. The research of this paper is the complete and analytic descriptions of all extension of singular linear Hamiltonian operators.The thesis is divided into three sections according to contents.Chapter 1. Preference, we introduce the main contents of this paper.Chapter 2. In this chapter, we consider the following linear Hamiltonian system: where a is a regular point, while b is singular, i.e., b=+∞; at least one of W(t) and Q(t) are not integrable near b; W(t) and Q(t) are 2n×2n Hermitian matrices and locally integrable in [a, b);J is the canonical symplectic matrix, i.e., In is n×n unit matrix, W(t)≥0 is a weight function.If the deficiency index of singular linear Hamiltonian operator h is(d, d), then Im(h*y,y) is a quadratic form of rank 2d. Using this property, a new characterization of the domains of self-adjoint extensions of h is obtained.Chapter 3. Based on the second chapter, we obtain the analytic descriptions of all self-adjiont extension of singular linear Hamiltonian operators in the direct sum spaces.