The Point Spectrum And Self-adjoint Boundary Conditions Of The Linear Hamiltonian Systems

The research of Hamiltonian systems is the core of the differential operator.lt originates from the fields of mathematical science, life science and so on, especially in many of the mathematical models of quantum mechanics and biological engineering. Although Hamiltonian systems produced by practical problems is nonlinear, we may linearize it so that we can accurately describe some properties of actual problems. The research of this paper is the linear Hamiltonian systems.The thesis extends conclusion of differential operator and the products of differ-ential operators, using similar methods to discuss the linear Hamiltonian system, and getting a similar conclusion. This paper majors in the relationship between the num-bers of square-integrable solutions of the linear Hamiltonian systems with real-values spectral parameter and point spectrum of linear Hamiltonian operators with arbitrary deficiency index, and differential operators Tk(L)(k=1,2,…,m;m∈N, m≥2) produced by Hamilton operator L(y), a necessary and sufficient condition for the self-adjointness of product operator of Tm(L)…T2(L)T1(L) is obtained.The thesis is divided into three sections according to contents.Chapter 1. Introduction, we introduce the main contents of this paper.Chapter 2. In this chapter, we consider the following linear Hamiltonian system: Jy'(t)= (λW(t)+H(t))y(t), t∈I,I=[a,b), (1.3.1λ) where a is a regular point, while b is singular, i.e., b=+∞; at least one of W(t) and H(t) is not integrable near b; W(t), and H(t) are 2n x 2n Hermitian matrices and locally integrable in [a,b);J is the canonical symplectic matrix, i.e., In is n x n unit matrix, W(t)> 0 is a weight function.If, for allλin an open interval I, there are d of linearly independent square-integrable solutions, then for each self-adjoint extension of the minimal operator, the point spectrum is nowhere dense in I. Chapter 3. Under the basic conditionsW(2n(m-1))(t) H(2n(m-1))(t) are locally integrable inⅠ, and W(t)≥0, (2.2) if y∈Lw2[a,b), Lm(y)∈Lw2[a,b) and together with y(2n(m-1))∈ACloc([a,b)) imply that Lk(y)∈Lw2[a, b) (k=1,2,…, m-1), we say that Lm(y) is partially separated in Lw2 [a, b).Under the assumption that the power Lm(y) of the Hamiltonian operator of order 2n on [a, b) is partially separated in Lw2[a, b), we present a new characteriza-tion of self-adjoint boundary conditions for differential operators T(Lm) generated by Lm(y). For m differential operators Tk(L)(k=1,2,…,m;m∈N,m≥2) associated with L(y), the self-adjoint of product operator Tm(L)…T2(L)T1(L) is investigated, and a necessary and sufficient condition for the self-adjoint of Tm(L)…T2(L)T1(L) is obtained.