Regular Approximations Of Spectra Of Singular Second-order Symmetric Linear Difference Equations

The study of fundamental theory of difference equations started in1960s. Difference equations are effective tools in the study of discrete models, and are one of very important kinds of equations in mathematics. Due to a wide application in many fields such as mathematical science, life science, social science, etc., especially the celestial mechanics, quantum mechanics and bio-engineering, the research of difference equations has now become a prosperous project.The study of spectral theory of difference equations has played an impor-tant role in both theory and practical applications. Spectral problems can be divided into two classifications. Those defined over finite closed intervals with well-behaved coefficients are called regular; otherwise they are called singular. In1964, Atkinson [1] first studied the discrete boundary value problems. In1978, Hinton and Lewis [2] studied oscillation theory, the limit-point or limit-circle criteria, and spectral distribution of self-adjoint extension of the minimal operator of second order difference equations. Shi and Chen studied spectral theory of second-order vector difference equations in1999[3]. In2006, Shi and Sun [4] studied eigenvalues of second-order difference equations with coupled boundary conditions. Recently, Shi and Sun [5] gave out a complete character-ization of self-adjoint extensions for second-order symmetric linear difference equation. For more results about difference equations, we refer to [6-12].A relatively complete theoretical system for regular spectral problems has been developed, including properties of eigenvalues, orthogonality of cigen-functions, expansion theorem, and so on [1,12]. Compared with the regular case, the study of singular spectral problems are more complex and difficult because the spectral set of a singular difference (differential) equations may contain some continuous spectra except for isolated spectral points. For a regular spectral problem, some good spectral properties can be expected, in-cluding that the spectral set is discrete. It’s natural to propose the question how to approximate the spectra of a singular spectral problem by the spectra of some regular spectral problems. A solution to this question is not only of theoretical significance but also it provides a method of computing the spectra of the singular spectral problems.The regular approximations of spectra of singular differential operators have been investigated deeply. And some good results have been obtained, including spectral inclusion, spectral exactness, and so on. In1993, Baily etc. studied the problem of regular approximations of spectra of singular second-order differential operators [13]. Based on the dichotomy of the limit point and limit circle cases, they constructed three general self-adjoint boundary conditions. They constructed self-adjoint extension domain of the minimal operator and induced regular self-adjoint operators corresponding to every boundary condition. They proved that the induced regular self-adjoint opera-tors are strong resolvent convergent to the given singular self-adjoint operator. Recently, Cui studied the regular approximations of spectra of singular Hamil-tonian systems with one singular endpoint. For more results about regular approximations of spectra of differential equations, we refer to [15-25] and some references cited therein.With a rapid development of information technology and the wide applica-tions of digital computers, many mathematical models described by difference equations have appeared. They have attracted much attention from many experts and scholars. Those results are about qualitative issues, boundary value problems, spectral analysis, theory of self-adjoint subspace and other theories of difference equations. For a symmetric linear differential equation, provided that the definiteness condition is satisfied, its maximal operator is well-defined and the minimal operator is a symmetric operator, i.e., a densely defined Hermitian operator, and its adjoint is equal to the maximal operator. Thus, one can employ the spectral theory of symmetric operators to study it. However, for a symmetric linear difference equation, the minimal operator is non-densely defined, and the maximal operator is not well defined as an operator. In fact, it is a multi-valued operator. So the theory of symmetric operators is not applicable in the study of difference equations. In order to solve this problem, Coddington, Lesch, Malamud and other scholars have suc-cessfully extended the concepts and some results of densely defined Hermitian operators to Hermitian subspaces. Then, Shi [26] extended the classical GKN theory to Hermitian subspace, and based on this, she with Sun gave out a complete characterization of self-adjoint extensions for second-order symmet-ric linear difference equation [5], which is the first study of self-adjoint subspace extensions of difference equations. Recently, Shi, Shao and Ren studied some spectral properties of self-adjoint subspaces [27-28], which has laid a founda-tion for the study of spectral problems about difference equations.With a short history of study on spectral theory of difference equations, despite many good results, the theoretical system has not been complete. As far as we know, regular approximations of spectra of singular difference equa-tions have not been studied. Consequently, the study of this problem is not only of significance in practical applications but also makes complete the spec-tral theory of difference equations.This thesis is devoted to the regular approximations of spectra of singular second-order symmetric linear difference equations with one singular endpoint and two singular endpoints, separately. We firstly introduce the concept of a core of a closed subspace. In addition, we give a sufficient condition of strong resolvent convergence for self-adjoint subspaces and a sufficient condition of spectral inclusion and spectral exactness. Later, based on the results about the self-adjoint subspace extensions given in [5], we construct proper induced regular self-adjoint subspacc extensions to obtain the realization of regular spectral approximation.This thesis is divided into three chapters.In Chapter1, some basic concepts and fundamental theory of subspaces and second-order symmetric linear difference equations are introduced. We firstly give some basic concepts, including spectral inclusion, spectral exact-ness, and strong resolvent convergence for self-adjoint subspaces and some re- lated results. Later, we introduce a concept of a core of a closed subspace. In addition, we give a sufficient condition of strong resolvent convergence for self-adjoint subspaces and a sufficient condition of spectral inclusion and spectral exactness. These results play a key role in the study of regular approximations of spectra of singular second-order symmetric linear difference equations.In Chapter2, we pay attention to the study of regular approximations of spectra of singular second-order symmetric linear difference equations with one singular endpoint in the limit-circle and limit-point cases, respectively. In Section2.2, characterizations of self-adjoint subspace extensions of the cor-responding minimal subspace of singular second-order symmetric linear dif-ference equations that arc in the limit-circle and limit-point case at the sole singular endpoint are first given. Then we construct proper induced regular self-adjoint subspace extensions in these two cases. In Section2.3, regular approximations of spectra of singular second-order symmetric linear difference equations with one singular endpoint in the limit-circle case arc studied. We construct some new self-adjoint subspaces and show that the products of the new self-adjoint subspaces and the graphs of projection operators are strongly resolvent convergent to a given self-adjoint subspacc extension. Further, ap-plying the Green functions of their resolvent, we prove that the products of the resolvent of the new self-adjoint subspaces and the graphs of projection operators are norm convergent to the resolvent of a given self-adjoint sub-space. Finally, applying the related results given in Section1.2, we show that the induced regular self-adjoint subspaces are spectral exact for the given self-adjoint subspace in this case. In Section2.4, we study regular approximations of spectra of singular second-order symmetric linear difference equations with one singular endpoint in the limit-point case. With a similar argument to that used in Section2.3, one can show that the products of new self-adjoint subspaces and the graphs of projection operators are strongly resolvent conver-gent to a given self-adjoint subspace Then we obtain that the induced regular self-adjoint subspaces are spectral inclusive for a given self-adjoint subspace. At the end, we give an example to illustrate the fact that the spectral exact-ness may fail in general in this case by induced regular self-adjoint subspace constructed in this chapter. In Chapter3, we study regular approximations of spectra of singular second-order symmetric linear difference equations with two singular end-points. In Section3.2, characterizations of self-adjoint subspace extensions of the corresponding minimal subspace are first given in all the cases. Then we construct proper induced regular self-adjoint subspace extensions in each of these cases. In Section3.3, we study regular approximations of spectra of singular second-order symmetric linear difference equations that arc in the limit-circle case at both endpoints. With a similar argument to that used in Section2.3, We construct new self-adjoint subspaces and show that the prod-ucts of the new self-adjoint subspaces and the graphs of projection operators are strongly resolvent convergent to a given self-adjoint subspace extension. Further, applying the Green functions of their resolvent, we show that the products of the resolvent of the new self-adjoint subspaces and the graphs of projection operators are norm convergent to the resolvent of a given self-adjoint subspace extension. Finally, by the related results given in Section1.2, we obtain that the induced regular self-adjoint subspaces are spectral exact for the self-adjoint subspace in this case. In Section3.4, we study regular approximations of spectra of singular second-order symmetric linear difference equations with at least one endpoints in the limit point case. With a similar argument to that used in Section2.3, one can show that the products of new self-adjoint subspaces and the graphs of projection operators arc strongly re-solvent convergent to a given self-adjoint subspace. Then we prove that the induced regular self-adjoint subspaces are spectral inclusive for a given self-adjoint subspace extension. At the end of this chapter, we give an example to illustrate the fact that the spectral exactness may fail in general in these cases by induced regular self-adjoint subspace constructed in this chapter.