奇异差分方程谱的正则逼近
发布时间：2018-07-31 15:22
【摘要】：无论是连续谱问题还是离散谱问题都可分为两类：一类是定义在有限闭区间上且系数具有良好性质的称为正则谱问题,否则称为奇异谱问题.正则谱问题的研究已经形成相对比较完善的理论体系.与正则谱问题相比,奇异谱问题的研究还不是那么全面,还有很多非常重要的问题有待研究.众所周知,一个正则谱问题的谱仅由特征值构成,而一个奇异谱问题的谱可能同时由本质谱和特征值构成[5,48,81,92],因此,研究起来比较复杂和困难.一个奇异谱问题的谱能否由一列正则谱问题的特征值来逼近呢?显然,研究奇异谱问题的正则逼近无论在理论上还是在实际应用上都具有重要意义.奇异微分算子谱的正则逼近问题已经得到了广泛和深入的研究并得到了许多很好的结果,包括谱包含和谱准确[6,7,16,52,77,78,91,95,96].在1993年,Bailey, Everitt和Weidmamnn [6]研究了奇异微分Sturm-Liouville问题谱的正则逼近.首先,对于任给的一个奇异Sturm-Liouville问题,他们构造了一列正则问题.然后证明了这列正则问题关于该奇异问题在极限圆型下是谱准确的和在极限点型下仅是谱包含的.此外,在极限点型下,当谱是下方有界时,给出了本质谱下方的谱准确的结果.随后,Stolz, Weidmann和Teschl [77,78,91,95,96]研究了一般的奇异常微分算子谱的正则逼近问题.除了得到了类似于[6]中的结果外,还得到了本质谱间隙内谱准确的结果.特别地,Brown, Greenberg和Marletta[16]对给定的奇异四阶对称微分算子,构造了一列正则问题,证明了当端点为正则或极限圆型时,该奇异问题本质谱下方第k个特征值恰是这列正则问题的第k个特征值的极限.此外,[6,39,101]对一个奇异微分Sturm-Liouville问题构造了一列正则问题,并且证明了当至少有一端点为极限点型时,该奇异问题本质谱下方第k个特征值恰是这列正则问题的第k个特征值的极限.随着信息技术的飞速发展和电子计算机的广泛应用,涌现出了越来越多的离散系统,并且吸引了大量学者对它们进行研究.对于正则差分方程基本理论的研究已经有很长的历史,并且它们的谱理论已经形成比较完善的理论体系,例如特征值、特征函数的正交性以及展开定理[1,5,14,15,35,48,50,67,68,70,81,90,92].而对于奇异差分方程,在1964年,Atkinson [5]最早对其进行了研究.随后,取得了一些重要进展[10,11,12,13,22,28,46,57,59,60,62,63,64,65,71,75,76,86,87,97].特别地,奇异对称二阶线性差分方程和奇异离散线性Hamilton系统引起了人们的很大的兴趣,并得到了许多好的结果(参考文献[19,20,21,23,47,48,60,64,65,69,71,79,82,83,84,85,86,87,88,102],以及它们的参考文献).在1995年,Jirari在文献[48]中对二阶Sturm-Liouville差分方程和正交多项式做了系统的研究.在2004年,陈景年和史玉明在文献[19]中建立了奇异二阶线性差分方程的极限圆型和极限点型的判定原理.同年,綦建刚和陈绍著对奇异离散Hamilton系统做了研究,给出了纯点谱的存在性和谱的下界的判定条件[60].在2006年,史玉明[71]对一端奇异离散Hamilton系统建立了Weyl-Titchmarsh理论.随后,她和任国静对奇异离散Hamilton系统的亏指数和确定性条件进行了研究,并在此基础上给出了其自伴子空间扩张的完全刻画[64,65].最近,郑召文在[102]中给出了奇异离散Hamilton系统在有界扰动下其最大和最小亏指数的不变性的结果.对于奇异对称二阶线性差分方程和奇异离散线性Hamilton系统,还有很多问题有待研究解决.本文我们主要想研究它们谱的正则逼近问题.显然地,该问题的研究在数值分析和应用中都非常重要.众所周知,对于一个对称线性微分方程,只要相关的确定性条件成立,它的最大算子是良好定义的和最小算子是个对称算子,即一个稠定的Hermite算子,并且最小算子的伴随等于最大算子.因此,人们可以利用对称算子的谱理论对其进行研究.然而对一个对称线性差分方程,其最小算子有可能是非稠定的并且最小和最大算子有可能是多值的.详细讨论可参考文献[64,72,75].因此,一般不能用对称算子的谱理论来研究奇异差分方程的谱问题.例如,经典的von Neumann理论及其推广理论[24,34,93,94]以及适用于对称算子的亏指数稳定性理论[9,30,36,51,53,54,55,98,100]对其都不在适用.随着对算子理论的深入研究(有关线性算子理论的书籍,可参考[2,17,34,40,41,45,49,58,61,66,89,93,94]),人们发现了越来越多的多值算子和非稠定算子.例如,不满足确定性条件的连续线性Hamilton系统生成的算子和一般的离散线性Hamilton系统生成的算子[55,64,65,75].为了研究这一类算子同时也是为了进一步完善算子理论,亟待建立多值算子和非稠定Hermite算子理论.幸运的是,这一大难题可以用线性子空间理论(线性关系)来解决.在1961年,Arens [4]最早对线性关系进行了研究.一个线性关系实际上是相应乘积空间中的一个子空间,因此显然包含多值和非稠定的算子.随后,Cod-dington, Dijksma, Hassi, Snoo和其他学者成功地将对称算子的一些概念和结果推广到了Hermite子空间[3,8,18,24,25,26,27,29,31,32,33,42,43,44].特别地,Coddington和他的合作者成功地将经典的有关对称算子的von Neu-mann理论推广到了Hermite子空间,证明了Hermite子空间具有自伴子空间扩张当且仅当其正负亏指数相等[24,25,26,27].紧接着,史玉明将经典的Glazman-Krein-Naimark理论推广到了Hermite子空间[72],并且在此基础上,她和她的合作者孙华清和任国静分别给出了二阶对称线性差分方程和一般的线性离散Hamilton系统在正则和奇异情形下的自伴扩张的完全刻画[75,65].随后,她与邵春梅和任国静研究了自伴子空间谱的性质[74].最近,在上面这些工作的基础上,我们研究了自伴子空间序列的预解收敛及谱逼近[73],其中给出了几个判定自伴子空间序列强预解收敛和依范数预解收敛的充要条件；以及建立了一些自伴子空间序列谱包含和谱准确的判定原理.这些结果为研究奇异差分方程谱的正则逼近问题奠定了基础.据我们所知,目前有关奇异差分方程谱的正则逼近的结果还较少.在本文中,我们将利用线性子空间理论来分别研究奇异二阶对称线性差分方程和奇异离散线性Hamilton系统谱的正则逼近问题.本文分为三章.第一章是预备知识.介绍线性子空间的一些基本概念和结果以及奇异二阶对称线性差分方程和离散线性Hamilton系统的一些结果.第二章主要考虑奇异二阶对称线性差分方程的谱的正则逼近.首先,对给定的相应最小子空间的一个自伴子空间扩张,给出其诱导的正则自伴子空间扩张.然后,证明了当每个端点为正则或极限圆型时,给定的自伴子空间扩张的第k个特征值恰是诱导的正则自伴子空间扩张的第k个特征值的极限.特别地,我们首次研究了特征值逼近的误差估计,并且在这种情形下利用方程的系数给出了特征值逼近的误差估计.当至少有一端点为极限点型时,对给定的相应最小子空间的一个自伴子空间扩张,我们首先来构造特定的诱导的正则自伴子空间扩张.然后,在这种情形下证明了新的诱导的正则自伴子空间扩张序列关于预先给定的自伴子空间扩张在其本质谱间隙内是谱准确的.此外,在这种情形下证明了,给定的自伴子空间扩张,当其谱下方有界时,其本质谱下方的第k个特征值恰是新构造的诱导的正则自伴子空间扩张的第k个特征值的极限.第三章考虑奇异离散线性Hamilton系统谱的正则逼近.首先,对任意给定的自伴子空间扩张,我们构造了其诱导的正则自伴子空间扩张.然后,研究了如何将一个真子区间上的基础空间乘积空间中的一个子空间延展为整个区间上的基础空间乘积空间中的子空间,即如何做零延展.这个问题在连续情形下很容易解决,而在离散情形下则非常困难.更多地,给出了延展后空间的谱性质的不变性.因此,作为直接推论,我们得到了诱导的正则自伴子空间扩张到最初Hilbert空间乘积空间的子空间的延展.然后,我们研究了奇异离散线性Hamilton系统谱的正则逼近.在极限圆型时,证明了诱导的正则自伴子空间扩张关于给定的自伴子空间扩张是谱准确的.更进一步地,在这种情形下证明了给定的自伴子空间扩张的第k个特征值恰是其诱导的正则自伴子空间扩张的第k个特征值的极限.此外,在这种情形下我们利用方程的系数首次给出了特征值逼近的误差估计.最后,在极限点型和中间亏指数型下得到了谱包含的结果.也许由于中间亏指数型研究起来非常复杂和困难,据我们所知,无论在连续情形下还是离散情形下,已有文献中目前还没有有关中间亏指数型下谱的正则逼近问题的研究结果.因此,本文是首次对离散Hamilton系统在中间亏指数型下谱的正则逼近问题做了研究并给出了谱包含的逼近结果.
[Abstract]:Both continuous and discrete spectral problems can be divided into two categories: one is a regular spectrum problem which is defined on the finite closed interval and the coefficient has good properties. Otherwise, the study of the regular spectrum problem has formed a relatively perfect theoretical system. It is not so comprehensive that there are many very important problems to be studied. As we all know, the spectrum of a regular spectral problem is composed of only the eigenvalues, and the spectrum of a singular spectrum problem may be composed of both the mass spectrum and the eigenvalue [5,48,81,92], so it is more complicated and difficult to study. It is obvious that the regular approximation of the singular spectrum problem is of great significance both in theory and in practical applications. The regular approximation problem of the singular differential operator spectrum has been widely and deeply studied and many good results have been obtained, including spectral inclusion and spectral exact [6,7,16. 52,77,78,91,95,96]., in 1993, Bailey, Everitt and Weidmamnn [6], studied the regular approximation of the spectrum of singular differential Sturm-Liouville problems. First, a regular problem was constructed for a singular Sturm-Liouville problem given to them. Then, it was proved that the regular question is about the spectral accuracy under the limit circle. And at the limit point type, it is only included in the spectrum. In addition, at the limit point type, when the spectrum is below the boundary, the exact results of the spectrum below the mass spectrum are given. Then, Stolz, Weidmann and Teschl [77,78,91,95,96] study the regular approximation problem of the general singular ordinary differential operator spectra. Besides the results similar to those in [6], the results are also obtained. The exact results of the internal spectrum of this mass spectrum are obtained. In particular, Brown, Greenberg and Marletta[16] construct a regular problem for a given singular four order symmetric differential operator. It is proved that when the endpoint is a regular or limit circle, the K eigenvalue below the mass spectrum of this singular problem is exactly the K eigenvalue of the regular problem. In addition, [6,39101] constructs a regular problem for a singular differential Sturm-Liouville problem and proves that when at least one endpoint is the limit point type, the K eigenvalue below the mass spectrum of the singular problem is just the limit of the K eigenvalue of the regular problem. Widely used, more and more discrete systems have emerged and attracted a lot of scholars to study them. The basic theory of the regular difference equation has a long history, and their spectral theory has formed a relatively perfect theoretical system, such as the eigenvalue, the orthogonality of the characteristic function and the expansion theorem [1,5, 14,15,35,48,50,67,68,70,81,90,92]., for the singular difference equation, in 1964, Atkinson [5] has been studied at the earliest. Then, some important progress, [10,11,12,13,22,28,46,57,59,60,62,63,64,65,71,75,76,86,87,97]. especially, singular symmetric two order linear difference equation and singular discrete linear Hamilton system, have been obtained. People have a lot of interest and get a lot of good results (reference [19,20,21,23,47,48,60,64,65,69,71,79,82,83,84,85,86,87,88102], and their references). In 1995, Jirari made a systematic study of the two order Sturm-Liouville difference equation and orthogonal multinomial equation in the literature [48]. In 2004, Chen Jingnian and Shi Yu The determination principle of limit circle type and limit point type of singular two order linear difference equation is established in document [19]. In the same year, Qijian gang and Chen Shaozhuo have studied the singular discrete Hamilton system. The existence of pure point spectrum and the criterion for determining the lower bounds of the spectrum are given [60]. in 2006, the singular discrete Hamilton system at one end of Shi Yuming [71] The Weyl-Titchmarsh theory is established. Then, the loss index and the deterministic condition of her and Mr. Ren state for the singular discrete Hamilton system are studied, and on this basis, the complete characterization of the self adjoint space expansion is given by Zheng Zhaowen. In [102], Zheng Zhaowen gives the maximum and maximum of the singular dispersion Hamilton system under bounded disturbance. There are many problems to be solved for the singular symmetric two order linear difference equations and singular discrete linear Hamilton systems. We mainly want to study the regular approximation problem of their spectra. Obviously, the study of this problem is very important in numerical analysis and application. A symmetric linear differential equation, as long as the relevant deterministic conditions are established, its maximum operator is well defined and the smallest operator is a symmetric operator, that is, a thickened Hermite operator, and the adjoint of the smallest operator is equal to the maximum operator. A symmetric linear difference equation, its minimum operator may be non thickening and the smallest and maximum operator may be multivalued. A reference [64,72,75]. is discussed in detail. Therefore, the spectral theory of the singular differential equation can not be studied by the spectral theory of symmetric operators. For example, the canonical von Neumann theory and its extension theory [24,34,93,9 4] and the loss index stability theory [9,30,36,51,53,54,55,98100] suitable for symmetric operators are not applicable to them. With the in-depth study of operator theory (books on linear operator theory, reference to [2,17,34,40,41,45,49,58,61,66,89,93,94]), more and more multivalued operators and non thickening operators are found. For example, not The operators generated by continuous linear Hamilton systems with deterministic conditions and the operator generated by the general discrete linear Hamilton system, [55,64,65,75]., in order to study this kind of operator and to further improve the operator theory, need to establish the multi value operator and the non thickening Hermite arithmetic subtheory. Fortunately, this big problem can be used. Linear subspace theory (linear relation) is solved. In 1961, Arens [4] was the first to study linear relations. A linear relation is actually a subspace in the corresponding product space, so it is obviously a multivalued and non thickening operator. Then, Cod-dington, Dijksma, Hassi, Snoo and other scholars succeed in the symmetry operator. Some concepts and results are generalized to the Hermite subspace [3,8,18,24,25,26,27,29,31,32,33,42,43,44]., and Coddington and his collaborators have successfully extended the classical von Neu-mann theory of symmetric operators to the Hermite subspace, proving that the Hermite subspace has a self adjoint subspace expansion when and only if its positive and negative losses are found. When the exponent is equal to [24,25,26,27]., Shi Yuming extends the classical Glazman-Krein-Naimark theory to the Hermite subspace [72], and on this basis, she and her collaborators Sun Huaqing and Ren Guojing give the two order symmetric linear difference equations and the general linear dispersion Hamilton system in regular and singular cases. After the complete portrayed of [75,65]., she studied the property [74]. of self companion space spectrum with Shao Chunmei and Ren state. On the basis of these work, we studied the presolution convergence and spectral approximation [73] of the self adjoint subspace sequence, and some of the strong presolutions of self adjoint subspace sequences are convergent and the norm presolution is given. The sufficient and necessary conditions for convergence; and the establishment of some principles for determining the spectral inclusion and spectral accuracy of self adjoint space sequences. These results have laid the foundation for the study of the regular approximation of the spectra of singular differential equations. As we know, there are fewer results on the regular approximation of the spectra of singular difference equations. In this paper, we will use the linearity. The regular approximation problems of singular two order symmetric linear difference equations and singular discrete linear Hamilton systems are studied by subspace theory. This paper is divided into three chapters. The first chapter is the preparatory knowledge. Some basic concepts and results of linear subspace are introduced, as well as a singular two order symmetric linear difference equation and discrete linear Hamilton system. Some results. The second chapter mainly considers the regular approximation of the spectra of singular two order symmetric linear difference equations. First, a self adjoint subspace expansion for a given corresponding minimal subspace is extended, and its induced regular self adjoint subspace expansion is given. Then, it is proved that a given self adjoint subspace expansion when each endpoint is regular or limit circular. The K eigenvalue is exactly the limit of the K eigenvalue of the induced regular self adjoint subspace expansion. In particular, we first study the error estimate of the eigenvalue approximation, and in this case we use the coefficient of the equation to estimate the error estimate of the eigenvalue approximation. A self adjoint subspace expansion of subspace, we first construct a specific induced regular self adjoint subspace expansion. Then, in this case, it is proved that the new induced regular self adjoint subspace expansion sequence is accurate in the intrinsic spectral gap of the pre given self adjoint subspace expansion. In addition, it is proved in this case. It is clear that, given the self adjoint subspace expansion, when its spectrum is bounded, the K eigenvalue below its mass spectrum is just the limit of the K eigenvalue of the newly constructed regular self adjoint space expansion. The third chapter considers the regular approximation of the spectrum of singular discrete linear Hamilton systems. First, for any given self adjoint subspace expansion, I We construct its induced regular self companion subspace expansion. Then, we study how to extend a subspace in the product space of the product space on a subspace of the subinterval into the subspace of the product space of the base space on the whole interval, that is, how to do the zero extension. It is more difficult. More, we give the invariance of the spectral properties of the space after extension. As a direct inference, we get the extension of the induced regular self adjoint subspace to the subspace of the original Hilbert space. Then, we study the regular approximation of the spectrum of the singular discrete linear Hamilton system. In this case, it is proved that the K eigenvalue of a given self adjoint subspace expansion is just the limit of the K eigenvalue of its induced regular self adjoint space expansion in this case. In addition, we are in this case. The error estimates of eigenvalue approximation are given for the first time by using the coefficient of the equation. Finally, the results of spectral inclusion are obtained under the limit point type and the intermediate loss index type. Perhaps because the intermediate loss index type is very complex and difficult to be studied, it is not yet available in the existing literature, as we know, in the continuous case and in the discrete case. This paper is the first time to study the regular approximation problem of the discrete Hamilton system under the intermediate loss index model and give the approximation results of the spectral inclusion.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.3
[Abstract]:Both continuous and discrete spectral problems can be divided into two categories: one is a regular spectrum problem which is defined on the finite closed interval and the coefficient has good properties. Otherwise, the study of the regular spectrum problem has formed a relatively perfect theoretical system. It is not so comprehensive that there are many very important problems to be studied. As we all know, the spectrum of a regular spectral problem is composed of only the eigenvalues, and the spectrum of a singular spectrum problem may be composed of both the mass spectrum and the eigenvalue [5,48,81,92], so it is more complicated and difficult to study. It is obvious that the regular approximation of the singular spectrum problem is of great significance both in theory and in practical applications. The regular approximation problem of the singular differential operator spectrum has been widely and deeply studied and many good results have been obtained, including spectral inclusion and spectral exact [6,7,16. 52,77,78,91,95,96]., in 1993, Bailey, Everitt and Weidmamnn [6], studied the regular approximation of the spectrum of singular differential Sturm-Liouville problems. First, a regular problem was constructed for a singular Sturm-Liouville problem given to them. Then, it was proved that the regular question is about the spectral accuracy under the limit circle. And at the limit point type, it is only included in the spectrum. In addition, at the limit point type, when the spectrum is below the boundary, the exact results of the spectrum below the mass spectrum are given. Then, Stolz, Weidmann and Teschl [77,78,91,95,96] study the regular approximation problem of the general singular ordinary differential operator spectra. Besides the results similar to those in [6], the results are also obtained. The exact results of the internal spectrum of this mass spectrum are obtained. In particular, Brown, Greenberg and Marletta[16] construct a regular problem for a given singular four order symmetric differential operator. It is proved that when the endpoint is a regular or limit circle, the K eigenvalue below the mass spectrum of this singular problem is exactly the K eigenvalue of the regular problem. In addition, [6,39101] constructs a regular problem for a singular differential Sturm-Liouville problem and proves that when at least one endpoint is the limit point type, the K eigenvalue below the mass spectrum of the singular problem is just the limit of the K eigenvalue of the regular problem. Widely used, more and more discrete systems have emerged and attracted a lot of scholars to study them. The basic theory of the regular difference equation has a long history, and their spectral theory has formed a relatively perfect theoretical system, such as the eigenvalue, the orthogonality of the characteristic function and the expansion theorem [1,5, 14,15,35,48,50,67,68,70,81,90,92]., for the singular difference equation, in 1964, Atkinson [5] has been studied at the earliest. Then, some important progress, [10,11,12,13,22,28,46,57,59,60,62,63,64,65,71,75,76,86,87,97]. especially, singular symmetric two order linear difference equation and singular discrete linear Hamilton system, have been obtained. People have a lot of interest and get a lot of good results (reference [19,20,21,23,47,48,60,64,65,69,71,79,82,83,84,85,86,87,88102], and their references). In 1995, Jirari made a systematic study of the two order Sturm-Liouville difference equation and orthogonal multinomial equation in the literature [48]. In 2004, Chen Jingnian and Shi Yu The determination principle of limit circle type and limit point type of singular two order linear difference equation is established in document [19]. In the same year, Qijian gang and Chen Shaozhuo have studied the singular discrete Hamilton system. The existence of pure point spectrum and the criterion for determining the lower bounds of the spectrum are given [60]. in 2006, the singular discrete Hamilton system at one end of Shi Yuming [71] The Weyl-Titchmarsh theory is established. Then, the loss index and the deterministic condition of her and Mr. Ren state for the singular discrete Hamilton system are studied, and on this basis, the complete characterization of the self adjoint space expansion is given by Zheng Zhaowen. In [102], Zheng Zhaowen gives the maximum and maximum of the singular dispersion Hamilton system under bounded disturbance. There are many problems to be solved for the singular symmetric two order linear difference equations and singular discrete linear Hamilton systems. We mainly want to study the regular approximation problem of their spectra. Obviously, the study of this problem is very important in numerical analysis and application. A symmetric linear differential equation, as long as the relevant deterministic conditions are established, its maximum operator is well defined and the smallest operator is a symmetric operator, that is, a thickened Hermite operator, and the adjoint of the smallest operator is equal to the maximum operator. A symmetric linear difference equation, its minimum operator may be non thickening and the smallest and maximum operator may be multivalued. A reference [64,72,75]. is discussed in detail. Therefore, the spectral theory of the singular differential equation can not be studied by the spectral theory of symmetric operators. For example, the canonical von Neumann theory and its extension theory [24,34,93,9 4] and the loss index stability theory [9,30,36,51,53,54,55,98100] suitable for symmetric operators are not applicable to them. With the in-depth study of operator theory (books on linear operator theory, reference to [2,17,34,40,41,45,49,58,61,66,89,93,94]), more and more multivalued operators and non thickening operators are found. For example, not The operators generated by continuous linear Hamilton systems with deterministic conditions and the operator generated by the general discrete linear Hamilton system, [55,64,65,75]., in order to study this kind of operator and to further improve the operator theory, need to establish the multi value operator and the non thickening Hermite arithmetic subtheory. Fortunately, this big problem can be used. Linear subspace theory (linear relation) is solved. In 1961, Arens [4] was the first to study linear relations. A linear relation is actually a subspace in the corresponding product space, so it is obviously a multivalued and non thickening operator. Then, Cod-dington, Dijksma, Hassi, Snoo and other scholars succeed in the symmetry operator. Some concepts and results are generalized to the Hermite subspace [3,8,18,24,25,26,27,29,31,32,33,42,43,44]., and Coddington and his collaborators have successfully extended the classical von Neu-mann theory of symmetric operators to the Hermite subspace, proving that the Hermite subspace has a self adjoint subspace expansion when and only if its positive and negative losses are found. When the exponent is equal to [24,25,26,27]., Shi Yuming extends the classical Glazman-Krein-Naimark theory to the Hermite subspace [72], and on this basis, she and her collaborators Sun Huaqing and Ren Guojing give the two order symmetric linear difference equations and the general linear dispersion Hamilton system in regular and singular cases. After the complete portrayed of [75,65]., she studied the property [74]. of self companion space spectrum with Shao Chunmei and Ren state. On the basis of these work, we studied the presolution convergence and spectral approximation [73] of the self adjoint subspace sequence, and some of the strong presolutions of self adjoint subspace sequences are convergent and the norm presolution is given. The sufficient and necessary conditions for convergence; and the establishment of some principles for determining the spectral inclusion and spectral accuracy of self adjoint space sequences. These results have laid the foundation for the study of the regular approximation of the spectra of singular differential equations. As we know, there are fewer results on the regular approximation of the spectra of singular difference equations. In this paper, we will use the linearity. The regular approximation problems of singular two order symmetric linear difference equations and singular discrete linear Hamilton systems are studied by subspace theory. This paper is divided into three chapters. The first chapter is the preparatory knowledge. Some basic concepts and results of linear subspace are introduced, as well as a singular two order symmetric linear difference equation and discrete linear Hamilton system. Some results. The second chapter mainly considers the regular approximation of the spectra of singular two order symmetric linear difference equations. First, a self adjoint subspace expansion for a given corresponding minimal subspace is extended, and its induced regular self adjoint subspace expansion is given. Then, it is proved that a given self adjoint subspace expansion when each endpoint is regular or limit circular. The K eigenvalue is exactly the limit of the K eigenvalue of the induced regular self adjoint subspace expansion. In particular, we first study the error estimate of the eigenvalue approximation, and in this case we use the coefficient of the equation to estimate the error estimate of the eigenvalue approximation. A self adjoint subspace expansion of subspace, we first construct a specific induced regular self adjoint subspace expansion. Then, in this case, it is proved that the new induced regular self adjoint subspace expansion sequence is accurate in the intrinsic spectral gap of the pre given self adjoint subspace expansion. In addition, it is proved in this case. It is clear that, given the self adjoint subspace expansion, when its spectrum is bounded, the K eigenvalue below its mass spectrum is just the limit of the K eigenvalue of the newly constructed regular self adjoint space expansion. The third chapter considers the regular approximation of the spectrum of singular discrete linear Hamilton systems. First, for any given self adjoint subspace expansion, I We construct its induced regular self companion subspace expansion. Then, we study how to extend a subspace in the product space of the product space on a subspace of the subinterval into the subspace of the product space of the base space on the whole interval, that is, how to do the zero extension. It is more difficult. More, we give the invariance of the spectral properties of the space after extension. As a direct inference, we get the extension of the induced regular self adjoint subspace to the subspace of the original Hilbert space. Then, we study the regular approximation of the spectrum of the singular discrete linear Hamilton system. In this case, it is proved that the K eigenvalue of a given self adjoint subspace expansion is just the limit of the K eigenvalue of its induced regular self adjoint space expansion in this case. In addition, we are in this case. The error estimates of eigenvalue approximation are given for the first time by using the coefficient of the equation. Finally, the results of spectral inclusion are obtained under the limit point type and the intermediate loss index type. Perhaps because the intermediate loss index type is very complex and difficult to be studied, it is not yet available in the existing literature, as we know, in the continuous case and in the discrete case. This paper is the first time to study the regular approximation problem of the discrete Hamilton system under the intermediate loss index model and give the approximation results of the spectral inclusion.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.3
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