| The study of fundamental theory of continuous Hamiltonian systems started in 1930s. All conservative physical processes can be described by mathematical models in the Hamiltonian sytems. So the fundamental theory of continuous Hamiltonian systems has become one of the most important part of nonlinear science, and has played an significant role in mathematical science, life science and other fields. In particular, it has wide applications in quantum mechanics and bioengineering (cf. [4,67] and the references cited therein).Spectral problems of continuous 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. The study of regular spectral problems has been developed into a complete theoretical system, such as the properties of eigenvalues, the orthogonality of the eigenfunctions, the expansion theorem. Rayleigh principle. Compared with regular case, the study of singular spectral problems are more complex and difficult because in the regular case, the spectral set is discrete, but in the singular case other spectrum such as essential spectrum may occur, except for eigenvalues.The study of self-adjoint extension of symmetric operators plarys an important role in the spectral theory. There are two methods to explore the self-adjoint extension of symmetric operators. The first method is von Neumann theory (cf. [99]). The classical von Neumann theory provides a sufficient and necessary condition for a closed symmetric operator in an abstract Hilbert space to have self-adjoint extension. The self-adjoint extensions of a closed symmetric operator can be derived from its adjoint operator by adding proper boundary conditions. The second method is Glazman-Krein-Naimark (GKN) theory (cf. [68]), which was founded by the three mathematician. Glazman, Krein and Naimark in 1950. The GKN theory points out that all self-adjoint operator extension can be obtained by adding proper boundary condition to GKN-sets. For continuous linear Hamiltonian systems, provided that the definiteness condition is satisfied, the maximal operator is well-defined and the minimal operator is densely defined. And then, we can show that the positive and negative defect indices are just equal to the number of the linear independent square integrable solutions in the upper and lower half-planes, respectively. (cf. [59,60,80]. And then by applying the classical von Neumann theory and GKN theory, all of the self-adjoint domains of continuous Hamiltonian systems, including higher order formally self-adjoint differential equations can be given (cf. [13,14,90,91,92], etc.). But, if the definiteness condition is not satisfied, then the maximal operator may be multiplied, that is it may not be an operator, and the minimal operator may be multi-valued and may be non-densely defined [64]. In this case, the methods mentioned above are not available.With the rapid development of information technology and the wide applications of digital computers, many mathematical models in discrete Hamiltonian sytems have ap-peared and been paid attention by more and more scholars (cf. [2,6,9.10,11,12,20,81] and their references). The appearance of discrete systems has a practical backgroud. As we know, continuous systems can be described by differential systems, while some systems (such as sample systems) can not be described by differential systems and can only be described by discrete systems. On the other hand, it is impossible to obtain an accurate solution of a general nonlinear differential system. So, we often compute its approximate solution by applying some discretization methods. In addition, discrete Hamiltonian systems originated from the discretization of continuous Hamiltonian sys-tems and from discrete processes acting in accordance with the Hamiltonian principle such as discrete physical problems, discrete control problems and so on. In spite of the similarity between the continuous and discrete Hamiltonian systems, there are many differences. Sometimes the results for continuous Hamiltonian systems are inconsistent with those for discrete one; sometimes the results on continuous and discrete systems are similar, but their corresponding methods are quite different.Spectral problems of discrete Hamiltonian systems can also be divided into two classifications. Those defined over finite intervals with well-behaved coefficients are called regular; otherwise they are called singular. Compared with continuous Hamilto-nian systems, some important problems for discrete Hamiltonian systems have not been studied. The study of regular spectral problems has a long history and many excellent results have been obtained (cf. [1,2,11,12,18,35,56,85]). For the singular prob-lems, Atkinson [6] first studied the second order difference equations over an infinite interval. His work was followed by Hinton and Lewis [48]. For the further research, Shi, Chen, Clark, Smith, Bohner, Dosly, Jirari, Sun and etc. gave their contributions in the spectral problems of second order and high order vector difference equations, as well as discrete Hamiltonian systems [16,19,54,66,71,72,84,85,86,87]. For singular discrete linear Hamiltonian systems Shi established its Titchmarsh-Weyl Theory [81]. Later, Sun, Shi and Chen [95] es-tablished its GKN theory. In 2007, Sun gave characterization of self-adjoint domain of the minimal operator, which is generated by the discrete linear Hamiltonian systems in her dissertation [89]. Subsequently, Shi and Sun found that even if the corresponding definiteness condition is satisfied, the maximal operator defined in [81,89,95] may be multi-valued, that is it is not well-defined as an operator, while the minimal operator may be non-densely defined. In addition, the minimal operator may be multi-valued. This is another important difference between difference equations and differential equa-tions.According to the extended von Neumann theory, a symmetric operator or a closed Hermitian subspace has a self-adjoint extension if and only if its positive and negative defect indices are equal. In addition, its self-adjoint extensions have a direct relationship with its defect index. So it is very important to explore the defect index not only for differential equations but also for difference equations. As we know, under certain definiteness condition, the positive and negative defect indices of 2m-th order formally self-adjoint differential equations with real coefficients are equal; that is, d+=d_=d. Moreover, d is equal to the number of linear independent square summable solutions of this equation with??E C\R. There many good results on the defect index of the differential operator, such as [27,28,29,30,33,37,42.55,57,58,62,63,65,70,98.102]. It is worthy to mention that Glazman showed in [43] that when the interval considered is (a, b)=(0,+?), where?=0 is a regular point, the defect index d satisfies the inequality m?d?2m, and all of the values in this range can be obtained. In addition, Mcleod in [65] gave an example of a fourth-order differential equation with complex coefficients for which the positive and negative defect indices are (2,3). For difference equations, the corresponding results are seldom.This dissertation is devoted to the spectral theory of discrete linear Hamiltonian systems, including definiteness conditions, defect indices, criteria for limit point case and limit circle case, characterizations of the self-adjoint subspace extensions and the self-adjoint operator extensions of discrete linear Hamiltonian systems. As a special case,2m-th order formally self-adjoint difference equations with will be considered. It is divided into five chapters.In Chapter 1, some basic concepts and useful results are introduced, including theory of linear subspaces, and the results on matriices and discrete linear Hamiltonian systems.In Chapter 2, we pay attention to the definiteness condition for discrete linear Hamiltonian systems. First, it is shown that the adjoint subspace of the minimal subspace is equal to the maximal subspace. This plays a key role in the study of the defect index and the self-adjoint extensions of the minimal subspace. Subsequently, the definiteness condition for discrete linear Hamiltonian systems and its several equivalent statements are given. In addition, three sufficient conditions for definiteness conditions are establishied. Based on these results, we establish the relationship between the defect index of the minimal subspace and the number of linearly independent square summable solutions. At the end of this chapter, several criteria for limit point case and limit circle case are obtained.Chapter 3 is devoted to self-adjoint subspacc extensions of the minimal subspace, which is generated by a discrete linear Hamiltonian system. The characterizations of the minimal and the maximal subspace are given. Based on these results, characteriza-tions of all the self-adjoint subspace extensions of a discrete linear Hamiltonian system are obtained in terms of boundary conditions and linear independent square summable solutions. As a consequence, the characterizations of all the self-adjoint subspace ex-tensions are given in the limit point case and limit circle case.A subspace has a self-adjoint operator extension only when it is an operator. More-over, all these self-adjoint operator extensions are included in its self-adjoint subspace extensions. So, all its self-adjoint operator extensions can be obtained by adding some conditions to its self-adjoint subspace extensions. In Chapter 4, some conditions for the minimal subspace to be an operator and to be densely defined are given respectively, and then some self-adjoint operator extensions of the minimal subspace are derived from its self-adjoint subspace extensions.In Chapter 5, we pay attention to 2mth order formally self-adjoint difference equa-tions. We show that when the coefficients are real and scalar, the defect index d satisfies the inequalities m?d?2m, and all values of in this range are realized. This parallels the above Glazman's result for differential equations. In addition, several criteria of the limit point and strong limit point cases are established. To end this chapter, the characterizations of all of the self-adjoint operator extensions of 2mth order formally self-adjoint difference equations are obtained. |
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