Spectral Theory For Linear Discrete Hamiltonian Systems

In 1835, Hamilton found the Hamiltonian principle, i.e., all the physical processes with negligible dissipation can be described by a suitable Hamiltonian formalism. Hamiltonian principle has become one of the most useful tools in research of physical sciences. Because of its wide use, the research of Hamiltonian systems has become a prosperous project.Study on spectral theory of linear Hamiltonian systems is of theoretical and practical significance. For example, Schrodinger equations are the elementary equations in quantum mechanics. There is a close link between behaviors of particles and wave functions of Schrodinger operators. The energy of a particle can be described by the spectra of Schrodinger operators, and the isolated points of spectrum are corresponding to the level of the energy of a particle, which explains how the particle shifts from one level to another level of the energy. This phenomenon can not be explained by classical mechanics. Moreover, there is a close relationship between the distribution of particles and the continuous spectrum. Schrodinger equations are special cases of Hamiltonian systems.The study of fundamental theory of continuous Hamiltonian systems has a long history (cf. [1, 2], and many references cited therein) and their spectral theory has also been investigated intensively and deeply. 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. For regular spectral problems, many good results have been obtained (cf. [3-11, 13]). Because spectra of singular differential operators may contain continuous spectrum except for the isolated spectral point, they can not be studied by only employing those methods used in studying spectral problems of bounded operators. Study on spectral theory for singular second-order symmetric differential operators (singular Sturm-Liouville theory) was started with Weyl's work in 1910 [14]. From then on, singular Sturm-Liouville theory provided one of the methods of studying the state of microcosmic particles in quantum mechanics, which just sprang up. So the singular Sturm-Liouville theory attracted a great deal of interest of mathematicians and physicists. Many well-known scholars, such as Titchmarsh, Coddington, Levinson, Weidmann, Hinton, and Krall, had developed Weyl's work and generalized it to Hamiltonian systems (cf. [3, 4, 6, 15-37], and some references cited therein). So, this theory is called the Weyl-Titchmarsh theory. Especially, Chaudhuri and Everitt [38] established a close relationship between the spectrum of the second-order self-adjoint differential operator and the analyticity of the Weyl function. Hinton and Shaw [39] extended their results to singular Hamiltonian differential systems. Therefore, one can study the spectra of these differential operators by investigating their Weyl functions.Identification of the self-adjoint extensions and characterizations of all the self-adjoint extensions of differential minimal 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. The GKN theory was established by Glazman, Krein, and Naimark, who were the most known scientists in Soviet. They studied self-adjoint extensions of minimal operators by using symplectic algebra (cf. [40-42], and references cited therein). Niessen and Zettl discussed the Friedrichs extensions of the second-order symmetric differential operators that are bounded from below [43]. Z. Cao gave a complete 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 [44-46]. J. Sun gave a complete and direct characterization of all the self-adjoint extensions for high-order symmetric differential operators with middle defect indices [47].Some phenomena in life sciences, physics, mechanics, economic sciences and so on, can only be described by discrete mathematical models. Now, computers have become important tools in many scientific fields. Computers can only deal with relations of nu- merical measures, which are discrete or discretized. So. discrete systems have attracted a great deal of interest in the last thirty years (cf. [4, 48-76]). Discrete Hamiltonian systems originated from the discretization of continuous Hamiltonian systems and from discrete processes acting in accordance with the Hamiltonian principle. They play an important role in the practical applications. In spite of the similarity between the theories of continuous and discrete Hamiltonian systems, there are many differences. For example, sometimes some results on continuous Hamiltonian systems are dissimilar to relevant results on discrete Hamiltonian systems; some results of continuous and discrete Hamiltonian systems are similar, but the methods of the proofs are dissimilar. So, study of discrete Hamiltonian systems is more difficult and challenging in some way.Spectral problems of discrete linear Hamiltonian systems can be also divided into two types: regular and singular problems. Those defined over finite closed intervals with well-behaved coefficients are called regular; otherwise they are called singular. Compared with continuous Hamiltonian systems, some important problems for discrete Hamiltonian systems have not been studied. For regular spectral problems of discrete Hamiltonian systems, Atkinson, Bohner, Shi and many others obtained some good results [4, 51, 66, 67, 70, 75]. The study of spectra for singular discrete Hamiltonian systems was started with Atkinson's work in 1964 [4]. This work was followed by Hinton, Lewis, Mingarelli, Jirari, Shi, Clark, and many others (cf. [61, 63, 62, 71, 57-59, 65, 73], and many references cited therein). Especially, Shi has established the Weyl-Titchmarsh fundamental theory for singular linear discrete Hamiltonian systems in [71]. A close relationship between the spectrum of the Hamiltonian operator in the limit point case and the analyticity of the Weyl function M(X) was established [71]. Therefore, one can study the spectrum of the Hamiltonian operator by investigating M(X) in the limit point case. Up to now, there are a few results about the characterization of all the self-adjoint extensions of linear discrete Hamiltonian operator. We have only found that the GKN theory for singular discrete Hamiltonian systems has been established by Sun and Shi [74]. A series of results on the self-adjoint extensions of the operators induced by the singular discrete Hamiltonian systems were obtained in [74].In this paper, we study spectral theory for linear discrete Hamiltonian systems. We establish several limit point criteria and strong limit point criteria for singular linear discrete linear Hamiltonian systems, study spectral distribution for the singular linear discrete Hamiltonian operators, give a characterization of all the self-adjoint extensions of the singular linear discrete Hamiltonian operators, and compare the eigenvalues of second-order regular difference equations with different coupled boundary conditions.The limit point case of continuous Hamiltonian systems can be further divided into two cases: the strong and the weak limit point cases (cf. [21-23, 35]). This classification is useful in studying the spectra of Hamiltonian operators. In Chapter 1, we shall make use of the sufficient and necessary conditions on limit point case established by using all the elements of the domain of maximal operator [71, Theorem 6.15] to give a further classification for the limit point case of discrete Hamiltonian systems: the strong and the weak limit point cases. In addition, we shall also give a definition of the strong and weak limit point cases for second-order vector difference equations.So far there have been few the limit point criteria established in terms of the coefficients of discrete Hamiltonian systems since the higher-dimensional systems are complex. In Chapter 2, several criteria of the limit point case are first established in terms of the coefficient matrices of discrete Hamiltonian systems in Section 1. In Section 2, a sufficient and necessary condition for the strong limit point case is obtained in terms of the elements of the domain of the maximal operator. Based on it, several sufficient conditions for the strong limit point case are obtained in terms of the coefficients of discrete Hamiltonian systems. As their applications, two criteria of the strong limit point case for the second-order vector equation are established.It has been known that one can study the spectra of continuous Hamiltonian operators and discrete Hamiltonian operators by investigating their Weyl functions. However, only a few results on the spectra of second order and higher-order singular scalar differential operators were obtained by using the singularity of the Weyl function since the Weyl function is difficult to study. In Chapter 3, we investigate spectra of Hamiltonian operators by using the Weyl function. In Section 3, the analyticity of the Weyl function M(λ) is first studied by the Schwarz reflection principle in the case that the system is in the limit point case. Based on it, some results about the spectrum are obtained in the strong limit point case. Especially, a sufficient condition for pure point spectrum is given. As their applications, the spectral analysis of second-order difference operators is studied finally.Chapter 4 is devoted to a complete and direct characterization of all the self-adjoint extensions of the discrete Hamiltonian minimal operator. We firstly decompose the domains of the relevant operators. Based on it, a complete and direct characterization of all the self-adjoint extensions in terms of the square summable solutions is given by using the GKN theory. These results provide a supplement to the theory of discrete Hamiltonian systems, and give a method to identify self-adjoint discrete Hamiltonian operators.The comparison between eigenvalues of the periodic and antiperiodic boundary value problems attracted a great deal of interest in 1950's, and some beautiful results were obtained [3]. Wang and Shi [75] compared the eigenvalues of the periodic and antiperiodic boundary value problems of second-order difference equations. They obtained beautiful inequalities among the eigenvalues, which are similar to those for second-order ordinary differential equations. In chapter 5, we compare the eigenvalues of second-order difference equations with different coupled boundary conditions, and obtain relationships among those eigenvalues. These results extend the main results obtained in [75]. We shall remark that Coddington and Levinson mainly used the Priifer transformation of the second-order differential equations in their proof. Though the discrete Priifer transformation has been established in [52], a similar method is difficultly employed in studying the discrete problem. Motivated by [75], we apply some results obtained by Shi and Chen [66] to prove the existence of eigenvalues and to calculate the number of these eigenvalues; and apply some oscillation results obtained by Atkinson [4] to compare the eigenvalues of second-order difference equations with different coupled boundary conditions.