Regular Approximation Of Spectra Of Singular Difference Equations

Either continuous or discrete spectral problems can be divided into two classifications. Those defined over finite closed intervals with well-behaved co-efficients are called regular; otherwise they are called singular. The spectral theory of regular problems has formed a relatively complete theoretical sys-tem. Compared with regular spectral problems, some important problems for singular spectral problems have not been studied.It is well known that the spectrum of a regular problem consists of eigen-values, while the spectrum of a singular problem may consist of essential spec-tral points as well as eigenvalues (cf. [5,48,81,92]). Thus, it is very difficult and complicated to study them. Can the spectrum of a singular problem be approximated by eigenvalues of a sequence of regular problems? Obviously, the study of regular approximation of spectra of singular spectral problems plays an important role in both theoretical and practical applications.Regular approximation of spectra of singular differential equations has been investigated widely and deeply, and some good results have been ob-tained including spectral inclusion and spectral exactness [6,7,16,52,77,78, 91,95,96]. In 1993, Bailey, Everitt, and Weidmamnn studied regular approx-imation of singular differential Sturm-Liouville problems [6]. Firstly, for any given singular Sturm-Liouville problem, they constructed a sequence of reg-ular problems. Then, they showed that this sequence of regular problems is spectrally exact for the singular problem in the limit circle case (briefly, l.c.c.) and only spectrally inclusive for it in the limit point case (briefly, l.p.c). In addition, spectral exactness below the essential spectrum was given under the condition that the spectrum is bounded below in l.p.c. Later, Stolz, Weidman-n, and Teschl [77,78,91,95,96] investigated regular approximation of spectra of general singular ordinary differential operators. They not only obtained the results analogous to those in [6] but also got spectral exactness for isolated eigenvalues in essential spectral gaps. In particular, Brown, Greenberg, and Marietta [16] constructed a sequence of regular problems for a given fourth-order singular symmetric differential operator and showed that the eigenvalues of the singular problem are exactly the limits of eigenvalues of this sequence in the case that each endpoint is either regular or in l.c.c. Moreover, the authors in [6,39,101] constructed a sequence of regular problems for a singular dif-ferential Sturm-Liouville problem and showed that the k-th eigenvalue below the essential spectrum of the singular problem is exactly the limit of the k-th eigenvalues of this sequence in the case when at least one endpoint is in l.p.c.With the development of information technology and the wide application-s of digital compute, more and more discrete systems have appeared and they have attracted a lot of attention. The study of fundamental theory of regular difference equations has a long history and their spectral theory has formed a relatively complete theoretical system such as eigenvalue problems, orthogonal-ity of eigenfunctions and expansion theory (cf., [1,5,14,15,35,48,50,67,68, 70,81,90,92]). Spectral problems for singular difference equations were first-ly studied by Atkinson [5] in 1964, and some significant progresses have been made since then (cf., [10,11,12,13,22,28,46,57,59,60,62,63,64,65,71,75, 76,86,87,97]). Especially, research on spectral theory of singular second-order symmetric linear difference equations and discrete Hamiltonian systems has at-tracted a great deal of interest and some good results have been obtained (cf., [19,20,21,23,47,48,60,64,65,69,71,79,82,83,84,85,86,87,88,102], and references cited therein). In 1995, Jirari [48] studied the second-order Sturm-Liouville difference equations and orthogonal polynomials. In [19], Chen and Shi established the limit circle and limit point criteria for second-order linear difference equations in 2004. In the same year, Qi and Chen studied for the sin- gulax discrete Hamiltonian system and gave the lower bound for the spectrum and the presence of pure point spectrum for it [60]. In 2006, Shi [71] established the Weyl-Titchmarsh theory for singular discrete linear Hamiltonian systems with one singular endpoint. Later, she with Ren studied the deficiency in-dices and definiteness conditions and gave out complete characterizations of self-adjoint subspace extensions for singular discrete linear Hamiltonian sys-tems [64,65]. Recently, Zheng in [102] obtained the invariance of the minimal and maximal deficiency indices under bounded perturbation for singular dis-crete linear Hamiltonian systems. For singular second-order symmetric linear difference equations and discrete linear Hamiltonian systems, there are many problems need to be researched and resolved. In this dissertation, we want to study regular approximation of their spectra. Obviously, the study for this problem is very important in numerical analysis and applications.It is well known that, for a symmetric linear differential equation, provided that the related definiteness condition is satisfied, its maximal operator is well-defined and its minimal operator is a symmetric operator, i.e., a densely defined Hermitian operator, whose 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, its minimal operator may not be densely defined, and its minimal and maximal operators may be multi-valued. We refer to Refs. [64,72,75] for detailed discussions. So one cannot apply the spectral theory of symmetric operators to study spectral problems for singular difference equations in general. For example, the classical von Neumann theory and its generalization (cf. [24,34,93,94]) and the theory of stability of deficiency indices for symmetric operators [9,30,36,51,53,54,55, 98,100] are not applicable to it.With further research of operator theory (For the theory of linear opera-tors, readers are referred to the books [2,17,34,40,41,45,49,58,61,66,89, 93,94]), more and more multi-valued operators and non-densely defined oper-ators have been found. For example, the operators generated by those linear continuous Hamiltonian systems, which do not satisfy the definiteness condi- tions, and general linear discrete Hamiltonian systems may be multi-valued or not densely defined in their corresponding Hilbert spaces (cf. [55,64,65,75]). Motivated by the need to consider the operators of this kind as well as en-richment of many aspects of operator theory, it is necessary and urgent to establish theory of multi-valued operators and non-densely defined Hermitian operators.Fortunately, this major difficulty can be overcome by using the theory of linear subspaces (i.e., linear relations). In 1961, Arens [4] initiated the study of linear relations. A linear relation is actually a subspace in a related product space, and obviously includes multi-valued and non-densely defined linear oper-ators in the related space. Later, Coddington, Dijksma, Hassi, Snoo, and other scholars successfully extended the concepts and some results of symmetric op-erators to Hermitian subspaces [3,8,18,24,25,26,27,29,31,32,33,42,43,44]. In particular, Coddington and his coauthors successfully extended the classi-cal von Neumann theory for symmetric operators to Hermitian spaces, and showed that a Hermitian subspace has a self-adjoint subspace extension if and only if its positive and negative deficiency indices are equal [24,25,26,27]. Later, Shi extended the classical Glazman-Krein-Naimark (briefly, GKN) the-ory to Hermitian subspace [72], and based on this, she with her coauthors Sun and Ren gave out complete characterizations of self-adjoint extensions for second-order symmetric linear difference equation and general linear discrete Hamiltonian systems in both regular and singular cases, separately [75,65]. Then, she studied some spectral properties of self-adjoint subspaces together with her coauthors Shao and Ren [74]. Recently, based on the above results, we studied the resolvent convergence and spectral approximation of sequences of self-adjoint subspaces [73], where several equivalent and sufficient conditions for convergence of sequences of self-adjoint subspaces in the strong and norm resolvent senses were given; and some criteria were established for spectral inclusion and spectral exactness of a sequence of self-adjoint subspaces. These results have laid a foundation for the study of regular approximation of spectra of singular difference equations.To the best of our knowledge, there seem a few results about regular approximation of spectra of singular symmetric linear difference equations. In the present paper, we shall apply the theory of linear subspace to study regular approximation of spectra for singular second-order symmetric linear difference equations and singular discrete linear Hamiltonian systems, respectively.This dissertation is divided into three chapters. In Chapter 1, some basic concepts and useful results are introduced, including theory of linear subspaces, and the results on singular second-order symmetric linear difference equations and discrete linear Hamiltonian systems.In Chapter 2, we pay attention to regular approximation of spectra of singular second-order symmetric linear difference equations. Firstly, for any given self-adjoint subspace extension of the corresponding minimal subspace, its induced regular self-adjoint subspace extensions are introduced. Then, it is shown that the k-th eigenvalue of the given self-adjoint subspace extension is exactly the limit of the k-th eigenvalues of the induced regular self-adjoint subspace extensions in the case that each endpoint is either regular or in l.c.c. In particular, we firstly investigate the error estimate for the approximation of eigenvalues and give the error estimate in terms of the coefficients of the equation in this case. In the case that at least one endpoint is in l.p.c, for any given self-adjoint subspace extension of the corresponding minimal subspace, some certain induced regular self-adjoint subspace extensions are constructed. Then, spectral exactness in every gap of the essential spectrum of any self-adjoint subspace extension is obtained in this case. In addition, in this case, it is shown that the k-th eigenvalue below the essential spectrum of any given self-adjoint subspace extension is exactly the limit of the k-th eigenvalues of the sequence of the new constructed induced regular self-adjoint subspace extensions under the condition that the spectrum is bounded below.Chapter 3 is devoted to regular approximation of spectra of singular dis-crete linear Hamiltonian systems. Firstly, the induced regular self-adjoint sub-space extensions for any given elf-adjoint subspace extension are constructed. Then, we pay attention to how to extend a subspace in the product space of the fundamental spaces on a proper subinterval to a subspace in that on the original interval, i.e., how to do the "zero extensions". This problem can be very easily solved in the continuous case but hard in the discrete case. Further, the invariance of spectral properties of the extended subspaces is given. As a consequence, the extension from the induced regular self-adjoint subspace extension to a subspace in the product space of the original Hilbert spaces is given, and the invariance of spectral properties of the extended subspaces is obtained. Then, regular approximation of spectra of singular discrete linear Hamiltonian systems is studied. It is shown that the sequence of induced regu-lar self-adjoint subspace extensions is spectrally exact for any given elf-adjoint subspace extension in the limit circle case. In addition, it is obtained that the k-th eigenvalue of any given self-adjoint subspace extension is exactly the limit of the k-th eigenvalues of the induced regular self-adjoint subspace extensions in this case. Furthermore, error estimates for the approximation of eigenvalues in terms of the coefficients of the system are firstly given in this case. Finally, it is shown that spectral inclusion holds in the limit point and the intermediate deficiency index cases. Maybe due to that the intermediate deficiency index case is very complicated and difficult to study, as far as we know, no matter in the continuous or discrete case, there have been no results about regular approximation of spectra of singular problems in the intermediate deficiency index case in the existing literature. Therefore, in this dissertation, we inves-tigate regular approximation of spectra of singular discrete linear Hamiltonian systems in the intermediate deficiency index case for the first time and give the result of spectral inclusion.