發(fā)布時(shí)間：2018-07-31 15:22

【摘要】：無論是連續(xù)譜問題還是離散譜問題都可分為兩類：一類是定義在有限閉區(qū)間上且系數(shù)具有良好性質(zhì)的稱為正則譜問題,否則稱為奇異譜問題.正則譜問題的研究已經(jīng)形成相對(duì)比較完善的理論體系.與正則譜問題相比,奇異譜問題的研究還不是那么全面,還有很多非常重要的問題有待研究.眾所周知,一個(gè)正則譜問題的譜僅由特征值構(gòu)成,而一個(gè)奇異譜問題的譜可能同時(shí)由本質(zhì)譜和特征值構(gòu)成[5,48,81,92],因此,研究起來比較復(fù)雜和困難.一個(gè)奇異譜問題的譜能否由一列正則譜問題的特征值來逼近呢?顯然,研究奇異譜問題的正則逼近無論在理論上還是在實(shí)際應(yīng)用上都具有重要意義.奇異微分算子譜的正則逼近問題已經(jīng)得到了廣泛和深入的研究并得到了許多很好的結(jié)果,包括譜包含和譜準(zhǔn)確[6,7,16,52,77,78,91,95,96].在1993年,Bailey, Everitt和Weidmamnn [6]研究了奇異微分Sturm-Liouville問題譜的正則逼近.首先,對(duì)于任給的一個(gè)奇異Sturm-Liouville問題,他們構(gòu)造了一列正則問題.然后證明了這列正則問題關(guān)于該奇異問題在極限圓型下是譜準(zhǔn)確的和在極限點(diǎn)型下僅是譜包含的.此外,在極限點(diǎn)型下,當(dāng)譜是下方有界時(shí),給出了本質(zhì)譜下方的譜準(zhǔn)確的結(jié)果.隨后,Stolz, Weidmann和Teschl [77,78,91,95,96]研究了一般的奇異常微分算子譜的正則逼近問題.除了得到了類似于[6]中的結(jié)果外,還得到了本質(zhì)譜間隙內(nèi)譜準(zhǔn)確的結(jié)果.特別地,Brown, Greenberg和Marletta[16]對(duì)給定的奇異四階對(duì)稱微分算子,構(gòu)造了一列正則問題,證明了當(dāng)端點(diǎn)為正則或極限圓型時(shí),該奇異問題本質(zhì)譜下方第k個(gè)特征值恰是這列正則問題的第k個(gè)特征值的極限.此外,[6,39,101]對(duì)一個(gè)奇異微分Sturm-Liouville問題構(gòu)造了一列正則問題,并且證明了當(dāng)至少有一端點(diǎn)為極限點(diǎn)型時(shí),該奇異問題本質(zhì)譜下方第k個(gè)特征值恰是這列正則問題的第k個(gè)特征值的極限.隨著信息技術(shù)的飛速發(fā)展和電子計(jì)算機(jī)的廣泛應(yīng)用,涌現(xiàn)出了越來越多的離散系統(tǒng),并且吸引了大量學(xué)者對(duì)它們進(jìn)行研究.對(duì)于正則差分方程基本理論的研究已經(jīng)有很長(zhǎng)的歷史,并且它們的譜理論已經(jīng)形成比較完善的理論體系,例如特征值、特征函數(shù)的正交性以及展開定理[1,5,14,15,35,48,50,67,68,70,81,90,92].而對(duì)于奇異差分方程,在1964年,Atkinson [5]最早對(duì)其進(jìn)行了研究.隨后,取得了一些重要進(jìn)展[10,11,12,13,22,28,46,57,59,60,62,63,64,65,71,75,76,86,87,97].特別地,奇異對(duì)稱二階線性差分方程和奇異離散線性Hamilton系統(tǒng)引起了人們的很大的興趣,并得到了許多好的結(jié)果(參考文獻(xiàn)[19,20,21,23,47,48,60,64,65,69,71,79,82,83,84,85,86,87,88,102],以及它們的參考文獻(xiàn)).在1995年,Jirari在文獻(xiàn)[48]中對(duì)二階Sturm-Liouville差分方程和正交多項(xiàng)式做了系統(tǒng)的研究.在2004年,陳景年和史玉明在文獻(xiàn)[19]中建立了奇異二階線性差分方程的極限圓型和極限點(diǎn)型的判定原理.同年,綦建剛和陳紹著對(duì)奇異離散Hamilton系統(tǒng)做了研究,給出了純點(diǎn)譜的存在性和譜的下界的判定條件[60].在2006年,史玉明[71]對(duì)一端奇異離散Hamilton系統(tǒng)建立了Weyl-Titchmarsh理論.隨后,她和任國(guó)靜對(duì)奇異離散Hamilton系統(tǒng)的虧指數(shù)和確定性條件進(jìn)行了研究,并在此基礎(chǔ)上給出了其自伴子空間擴(kuò)張的完全刻畫[64,65].最近,鄭召文在[102]中給出了奇異離散Hamilton系統(tǒng)在有界擾動(dòng)下其最大和最小虧指數(shù)的不變性的結(jié)果.對(duì)于奇異對(duì)稱二階線性差分方程和奇異離散線性Hamilton系統(tǒng),還有很多問題有待研究解決.本文我們主要想研究它們譜的正則逼近問題.顯然地,該問題的研究在數(shù)值分析和應(yīng)用中都非常重要.眾所周知,對(duì)于一個(gè)對(duì)稱線性微分方程,只要相關(guān)的確定性條件成立,它的最大算子是良好定義的和最小算子是個(gè)對(duì)稱算子,即一個(gè)稠定的Hermite算子,并且最小算子的伴隨等于最大算子.因此,人們可以利用對(duì)稱算子的譜理論對(duì)其進(jìn)行研究.然而對(duì)一個(gè)對(duì)稱線性差分方程,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.詳細(xì)討論可參考文獻(xiàn)[64,72,75].因此,一般不能用對(duì)稱算子的譜理論來研究奇異差分方程的譜問題.例如,經(jīng)典的von Neumann理論及其推廣理論[24,34,93,94]以及適用于對(duì)稱算子的虧指數(shù)穩(wěn)定性理論[9,30,36,51,53,54,55,98,100]對(duì)其都不在適用.隨著對(duì)算子理論的深入研究(有關(guān)線性算子理論的書籍,可參考[2,17,34,40,41,45,49,58,61,66,89,93,94]),人們發(fā)現(xiàn)了越來越多的多值算子和非稠定算子.例如,不滿足確定性條件的連續(xù)線性Hamilton系統(tǒng)生成的算子和一般的離散線性Hamilton系統(tǒng)生成的算子[55,64,65,75].為了研究這一類算子同時(shí)也是為了進(jìn)一步完善算子理論,亟待建立多值算子和非稠定Hermite算子理論.幸運(yùn)的是,這一大難題可以用線性子空間理論(線性關(guān)系)來解決.在1961年,Arens [4]最早對(duì)線性關(guān)系進(jìn)行了研究.一個(gè)線性關(guān)系實(shí)際上是相應(yīng)乘積空間中的一個(gè)子空間,因此顯然包含多值和非稠定的算子.隨后,Cod-dington, Dijksma, Hassi, Snoo和其他學(xué)者成功地將對(duì)稱算子的一些概念和結(jié)果推廣到了Hermite子空間[3,8,18,24,25,26,27,29,31,32,33,42,43,44].特別地,Coddington和他的合作者成功地將經(jīng)典的有關(guān)對(duì)稱算子的von Neu-mann理論推廣到了Hermite子空間,證明了Hermite子空間具有自伴子空間擴(kuò)張當(dāng)且僅當(dāng)其正負(fù)虧指數(shù)相等[24,25,26,27].緊接著,史玉明將經(jīng)典的Glazman-Krein-Naimark理論推廣到了Hermite子空間[72],并且在此基礎(chǔ)上,她和她的合作者孫華清和任國(guó)靜分別給出了二階對(duì)稱線性差分方程和一般的線性離散Hamilton系統(tǒng)在正則和奇異情形下的自伴擴(kuò)張的完全刻畫[75,65].隨后,她與邵春梅和任國(guó)靜研究了自伴子空間譜的性質(zhì)[74].最近,在上面這些工作的基礎(chǔ)上,我們研究了自伴子空間序列的預(yù)解收斂及譜逼近[73],其中給出了幾個(gè)判定自伴子空間序列強(qiáng)預(yù)解收斂和依范數(shù)預(yù)解收斂的充要條件；以及建立了一些自伴子空間序列譜包含和譜準(zhǔn)確的判定原理.這些結(jié)果為研究奇異差分方程譜的正則逼近問題奠定了基礎(chǔ).據(jù)我們所知,目前有關(guān)奇異差分方程譜的正則逼近的結(jié)果還較少.在本文中,我們將利用線性子空間理論來分別研究奇異二階對(duì)稱線性差分方程和奇異離散線性Hamilton系統(tǒng)譜的正則逼近問題.本文分為三章.第一章是預(yù)備知識(shí).介紹線性子空間的一些基本概念和結(jié)果以及奇異二階對(duì)稱線性差分方程和離散線性Hamilton系統(tǒng)的一些結(jié)果.第二章主要考慮奇異二階對(duì)稱線性差分方程的譜的正則逼近.首先,對(duì)給定的相應(yīng)最小子空間的一個(gè)自伴子空間擴(kuò)張,給出其誘導(dǎo)的正則自伴子空間擴(kuò)張.然后,證明了當(dāng)每個(gè)端點(diǎn)為正則或極限圓型時(shí),給定的自伴子空間擴(kuò)張的第k個(gè)特征值恰是誘導(dǎo)的正則自伴子空間擴(kuò)張的第k個(gè)特征值的極限.特別地,我們首次研究了特征值逼近的誤差估計(jì),并且在這種情形下利用方程的系數(shù)給出了特征值逼近的誤差估計(jì).當(dāng)至少有一端點(diǎn)為極限點(diǎn)型時(shí),對(duì)給定的相應(yīng)最小子空間的一個(gè)自伴子空間擴(kuò)張,我們首先來構(gòu)造特定的誘導(dǎo)的正則自伴子空間擴(kuò)張.然后,在這種情形下證明了新的誘導(dǎo)的正則自伴子空間擴(kuò)張序列關(guān)于預(yù)先給定的自伴子空間擴(kuò)張?jiān)谄浔举|(zhì)譜間隙內(nèi)是譜準(zhǔn)確的.此外,在這種情形下證明了,給定的自伴子空間擴(kuò)張,當(dāng)其譜下方有界時(shí),其本質(zhì)譜下方的第k個(gè)特征值恰是新構(gòu)造的誘導(dǎo)的正則自伴子空間擴(kuò)張的第k個(gè)特征值的極限.第三章考慮奇異離散線性Hamilton系統(tǒng)譜的正則逼近.首先,對(duì)任意給定的自伴子空間擴(kuò)張,我們構(gòu)造了其誘導(dǎo)的正則自伴子空間擴(kuò)張.然后,研究了如何將一個(gè)真子區(qū)間上的基礎(chǔ)空間乘積空間中的一個(gè)子空間延展為整個(gè)區(qū)間上的基礎(chǔ)空間乘積空間中的子空間,即如何做零延展.這個(gè)問題在連續(xù)情形下很容易解決,而在離散情形下則非常困難.更多地,給出了延展后空間的譜性質(zhì)的不變性.因此,作為直接推論,我們得到了誘導(dǎo)的正則自伴子空間擴(kuò)張到最初Hilbert空間乘積空間的子空間的延展.然后,我們研究了奇異離散線性Hamilton系統(tǒng)譜的正則逼近.在極限圓型時(shí),證明了誘導(dǎo)的正則自伴子空間擴(kuò)張關(guān)于給定的自伴子空間擴(kuò)張是譜準(zhǔn)確的.更進(jìn)一步地,在這種情形下證明了給定的自伴子空間擴(kuò)張的第k個(gè)特征值恰是其誘導(dǎo)的正則自伴子空間擴(kuò)張的第k個(gè)特征值的極限.此外,在這種情形下我們利用方程的系數(shù)首次給出了特征值逼近的誤差估計(jì).最后,在極限點(diǎn)型和中間虧指數(shù)型下得到了譜包含的結(jié)果.也許由于中間虧指數(shù)型研究起來非常復(fù)雜和困難,據(jù)我們所知,無論在連續(xù)情形下還是離散情形下,已有文獻(xiàn)中目前還沒有有關(guān)中間虧指數(shù)型下譜的正則逼近問題的研究結(jié)果.因此,本文是首次對(duì)離散Hamilton系統(tǒng)在中間虧指數(shù)型下譜的正則逼近問題做了研究并給出了譜包含的逼近結(jié)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.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175.3

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