Regular Spectral Approximations Of Singular Linear Differential Hamiltonian Operators

In 1835, Hamilton found the Hamiltonian principle, i.e., all the physical processes with negligible dissipation can be described by a suitable Hamil-tonian formalism. The Hamiltonian principle has always been one of most important tools in the field of nonlinear sciences. Because it is widely ap-plied in Quantum Mechanics, aeroplane science, life science and many other mathematical models, the research of Hamiltonian systems has now become a prosperous project. In the study of nonlinear Hamiltonian systems, one needs to investigate properties of their solutions close to the equilibrium points or periodic solutions. The linearization of Hamiltonian system is necessary. It becomes important to study the spectral theory of linear Hamiltonian sys-tems. The study on spectral theory of linear Hamiltonian systems is of both theoretical and practical significance.The study of fundamental theory of differential Hamiltonian systems has a long history(cf. [1,2]) and their spectral theory has also been investigated deeply. Spectral problems of linear differential Hamiltonian systems can be divided into two classifications. Those defined over finite closed intervals with well-behaved coefficients are called regular; otherwise they are called singular. It is difficult to discuss Singular spectral problems because the spectrum of a singular differential operator may contain some continuous spectrum except for isolated spectral points. Study on spectral theory for singular differential operators was started with Weyl's work of giving a dichotomy of the limit point and limit circle cases by introducing a m(λ) function for singular scalar second-order symmetric differential equations in 1910 [3]. This work has been developed and generalized to Hamiltonian differential systems by Titchmarsh, Coddington, Levinson, Weidmann, Hinton, Krall, Chen, Shi, Qi (cf.[4-29]). So, this theory is called the Weyl-Titchmarsh theory. Especially, Chaudhuri and Everitt established a close relationship between the spectrum of the second-order self-adjoint differential operator and the analyticity of the Weyl function [30]. Hinton and Shaw extended their results to a class of singular Hamiltonian differential systems [31]. Y. Shi investigated the exact relationship between the rank of the matrix radius and the number of square integrable solutions, and proved that the defect index of the corresponding minimal operator can be represented in terms of the rank of the matrix radius of the related limit set [26]. Therefore, one can study the defect indices and spectra of these differential operators by studying their Weyl functions.Self-adjoint extension is one of fundamental problems in study of spectral problems for linear ordinary differential operators. A symmetric operator has self-adjoint extensions if and only if its positive and negative defect indices are equal. It is a natural and important question how to give all its self-adjoint extension domains. There are many methods to describe the self-adjoint ex-tension domain such as the classical von Neumann theory. For a differential operator, the GKN theory established by Glazman, Krein, and Naimark, pro-vided a complete characterization of all its self-adjoint extensions in terms of GKN set. In the classical Titchmarsh-Weyl theory, some self-adjoint extension domains of a second-order singular symmetric differential operator are deter-mined by Weyl's solutions. Since these extension domains are constructed by separated boundary conditions, they do not include all the self-adjoint ex-tension domains of the operator. Based on the GKN theory, Z. Cao gave a characterization of all the self-adjoint extensions of symmetric second-order and high-order differential expressions with maximum defect indices in terms of square integrable solutions [32-34]. Later, J. Sun gave a complete character-ization of self-adjoint extensions of all high-order differential expressions with middle defect indices in terms of square integrable solutions [35]. Recently, H. Sun and Y. Shi gave a complete characterization of self-adjoint extensions of singular Hamiltonian systems [36,37].The spectrum of a singular differential operator may contain continuous spectrum except for isolated spectral points. For a regular spectral problem, some good spectral properties can be expected, including that the spectral set is discrete. It's natural to propose the question how to approximate a singular spectral problem with regular spectral problems. A solution to this question is not only of theoretical significance but also it provides a way of computing the spectrum of the singular differential operators. In 1993, Baily etc. studied regular spectral approximation problem of singular second-order differential operators [38]. Based on the dichotomy of the limit point and limit circle cases, they constructed three general self-adjoint boundary conditions. They constructed induced regular self-adjoint operator corresponding to every sin-gular boundary condition. They proved that the induced regular self-adjoint operators are strong resolvent convergent to the singular self-adjoint opera-tor. Later, Brown etc. studied the regular spectral approximation problem of singular fourth-order differential operators. But they only considered this problem with separated boundary conditions. In this paper, we study the regular spectral approximation of singular Hamiltonian systems with one sin-gular endpoint in the limit circle and limit point cases. Based on the results about the self-adjoint domains of Hamiltonian systems given in [36], we con-struct proper induced operators to obtain the realization of regular spectral approximation.This paper is divided into two chapters. In Chapter 1, we introduce the fundamental theory of linear operators and some basic results of linear dif-ferential Hamiltonian systems. In Section 1.2, we give some basic concepts, including defect index, strong resolvent convergence, and norm resolvent con-vergence, and investigate some properties of general self-adjoint operators. We show that if the product of self-adjoint operators and projection operators is strongly resolvent convergent, then the self-adjoint operators is spectral in-cluded. Furthermore, if the products of the resolvents of self-adjoint operators and projection operators are norm convergent, then the self-adjoint operators is spectral exact. These results provide a basis for studying the spectral ex-actness for linear singular Hamiltonian systems in the limit circle case. In Section 1.3, we introduce the maximal operator H and the minimal operator H0 associated with the differential Hamiltonian system with one singular end-point and give some of their properties. In addition, we give some properties of solutions of linear Hamiltonian systems.In Chapter 2, we study regular spectral approximations of linear differ-ential Hamiltonian systems with one singular endpoint in the limit circle and limit point cases. In Section 2.1, we point out that general linear differential Hamiltonian systems contain 2n-order linear symmetric differential equations. In Section 2.2, characterizations of self-adjoint extensions of the minimal op-erator Ho are first given in the limit point case and limit circle case. Then, we construct suitable induced regular self-adjoint operators in these two limit types. In Section 2.3, we study regular spectral approximations of linear dif-ferential Hamiltonian systems in the limit circle case. Applying the domains of self-adjoint extensions characterized by GKN sets, we show that the prod-ucts of the induced regular self-adjoint operators and projection operators are strongly resolvent convergent to a given self-adjoint operator 5. Further, ap-plying the Green formulas of their resolvents, we prove that the products of the resolvents of the induced regular self-adjoint operators and projection opera-tors are norm convergent to the resolvent of S. By the related results in Section 1.2, we prove that the induced regular self-adjoint operators are spectral exact for the self-adjoint operator S. In Section 2.4, we study regular spectral ap-proximations of linear differential Hamiltonian systems in the limit point case. With a similar argument to that used in Section 2.3, one can show that the product of the induced regular self-adjoint operators and projection operators is strongly resolvent convergent to a given self-adjoint operator S. Then the induced regular self-adjoint operators are spectral inclusive for S. In addition, under a certain condition, we construct a class of special regular self-adjoint domains and show that these regular self-adjoint operators are spectral exact for S in those intervals containing no essential spectrum of operator S. Finally, we apply our regular approximation results obtained in the above sections to 2n-order singular linear symmetric differential operators in Section 2.5.