常微分算子理论的发展
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摘要
常微分算子理论是以量子力学为应用背景,综合常微分方程、泛函分析、算子代数及空间理论等理论、方法发展起来的一门系统的、内容广泛的数学分支.它是解决数学物理方程以及大量科学技术应用问题的重要数学工具.常微分算子理论所研究的主要内容包括：自共轭域、谱分析、亏指数及逆谱问题等.
本文在查阅了大量的原始文献和有关研究文献的基础上,利用文献分析研究与文献比较研究的方法,从以下几个方面较系统地研究了常微分算子理论的发展历程.
一、通过对Sturm和Liouville的工作及其它关于记载这些成果的史料进行分析与研究,从以下几个方面探寻了常微分算子理论的源流：(1)Sturm和Liouville成果的研究背景；(2)分析Sturm和Liouville的工作；(3) Sturm-Liouville理论的意义；(4) Sturm和Liouville工作的后续发展.
二、通过对20世纪早期的一些关于二阶奇异边值问题的文献进行系统分析与考察,从以下几个方面论述了Weyl(1910), Dixon (1912) Stone (1932)和Titchmarsh (1940-1950)的工作对常微分算子理论发展的贡献.我们发现Weyl和Titchmarsh的成果基本上源于经典的实分析和复分析,而Stone的研究工作是Hilbert函数空间抽象理论中自共轭算子与线性常微分方程理论结合的产物.
1.1910年,Weyl不仅开创了奇异S-L微分方程的研究,而且首次考虑了微分方程的分析特征.特别是一些新概念和新成果的提出,使S-L理论在20世纪的发展步入了一个新的发展阶段,也为后来的von Neumann和Stone在微分算子理论方面的研究以及为Titchmarsh应用复变换技巧提供了思想渊源.
2.1912年,Dixon第一次将系数函数p,q,w的连续性条件由Lebesgue可积条件来代替,此Lebesgue可积性条件也是现代微分算子研究中对系数要求最低的条件.
3.1932年,Stone首次在Hilbert函数空问上讨论具有Lebesgue可积系数的二阶微分算子的一般理论.
4. Titchmarsh应用单个复变量函数的展开理论研究了正则情形和奇异情形的S-L边值问题.
三、通过分析与研究关于常微分算子自伴域描述的已有成果,系统地总结了常型和奇异常微分算子自伴域描述的发展脉络.
1.高阶常型微分算子自伴域的描述问题于20世纪50年代彻底解决,1954年Coddington利用矩阵理论和共轭边条件的有关结论,给出了以边条件形式表示的自伴域,这是一个直接的描述结果；同年,Naimark给出了拟微分算子自伴域的描述；1962年,Everitt用微分方程的线性独立解来描述算子的自伴域,在系数足够光滑的条件下,这三个结论是等价的.
2.通过分析奇异微分算子自伴域描述的一些重要成果,比如,Weyl-Titchmarsh自伴域,Everitt自伴域,曹之江-自伴域和孙炯-自伴域,论述了曹之江-自伴域的重要性,它是一种直接而完全的自伴域描述,使得奇异微分算子自伴域描述的问题彻底解决.
四、通过分析和考察大量的关于谱分析方面的文章,主要以离散谱和本质谱的判别为核心梳理了实自伴微分算子,加权的奇异微分算子及J-自伴微分算子离散谱的判别工作和几类特定微分算子本质谱的判别结果.
五、通过挖掘和考察大量的关于亏指数方面的第一手文献,系统地论述了奇异实对称微分算子和复对称微分算子在二阶和高阶情形下极限点型和圆型的判别工作
The theory of ordinary differential operators is playing an important role in the modern quantum mechanic, and is also an important facility in mathematical-physical equations and other applied technology fields. It is a systematic and comprehensive mathematical branch, which includes theories and methods in ordinary differential equations, functional analysis, operator algebra and function spaces. Some principal problems, including description of self-adjoint domains, analysis of spectrum, deficiency index, and inverse spectrum problemetc, are investigated in the theory of ordinary differential operators.
Based on the intensive reading of original sources and research literatures on the subject in consideration, this dissertation investigates systematically and thoroughly into the development history of the theory of ordinary differential operators by using the method of source analysis and the comparative approach of the literature. The main results of this paper are as follows:
First, Based on the brief introduction of the well-known Sturm and Liouville's results, the author systematically and thoroughly discuss the origins of the ordinary differential operators theory. The main contents of this part are as follows:(1) the origins of the Sturm-Liouville theory; (2) the work of Sturm and the work of Liouville; (3) the influence of the Sturm-Liouville theory;(4) the later development of the Sturm-Liouville theory.
Second, By reviewing carefully the early work about the singular Sturm-Liouville boundary value problems, which took place during the years from 1900 to 1950, in this chapterⅢthere are detailed discussions of contributions to from:Weyl(1910), Dixon(1912), Stone (1932) and Titchmarsh (1940-1950). The results of Weyl and Titchmarsh are essentially derived within classical, real and complex mathematical analysis. The results of Stone apply to examples of self-adjoint operators in the abstract theory of Hilbert spaces and in the theory of ordinary linear differential equations. The main contents of this part are as follow:
1. In 1910, the results of Weyl are the first to consider the singular case of the S-L differential equations it is the first structured consideration of the analytical properties of the equation. The rang of new definitions and results is remarkable and set the stage for the full development of S-L theory in the 20th century, as to be seen in the later theory of differential operators in the work of von Neumann and Stone, and in the application of complex variable techniques by Titchmarsh.
2. In 1912, the paper of Dixon seems to be the first paper in which the continuity conditions on the coefficientsp,q,ware replaced by the Lebesgue integrability conditions; these latter conditions are the minimal conditions to be satisfied by p,q,w within the environment given by the Lebesgue integral.
3. In 1932, the results of Stone seems to be the first extended account of the properties of S-L differential operators in the Hilbert function spaces, under the Lebesgue minimal conditions on the coefficients of differential equation.
4. Titchmarsh considered both the regular and singular case of S-L problems by applying the extensive theory of functions of a single complex variable.
Third, By analying the literatures in the past on characterization of self-adjoint domains, the author discusses the important results in the development of the theory of self-adjoint domains. The main content of this part is as follow:
1. In the regular case, in 1954, Coddington used the matrix theory and the conjugate boundary conditions, and given the description of self-adjoint domains of even order symmetric expressions. Naimark considered a symmetric quasi-differential expressions with two regular the boundary conditions. When coefficients are sufficiently smooth, Coddington's condition and Naimark's condition were proved equivalant. In 1962, Everitt given a result using linearly independent solutions of differential equation. This result is similar to Coddington's, and above conditions and Everitt's are equivalent.
2. In the singular case, by studying deeply into the Titchmarsh's result Everitt's result, cao zhi Jiang's result and sun jiong's result, it is pointed out that cao zhi Jiang's result is a complete and direct description of self-adjoint domains with the ones exposed by Everitt as special.
Fourth, based on the analysis of a number of papers on spectra analysis of ordinary differential operators, in particular, the author generalize the important criteria of the discrete spectrum of the real self-adjoint differential operators, weighted differential operators and J- self-adjoint differential operators, and of the essential spectrum of the classes of differential operators.
Fifth, through deep study on the first-hand information, the author systematically and thoroughly discuss the development on the deficiency indices theory for singular symmetric differential operators. In particular two important criterias were drawn as the core of the deficiency indices theory: the limit-point case and the limit-circle case of 2-nd order or higher order differential operators.
本文在查阅了大量的原始文献和有关研究文献的基础上,利用文献分析研究与文献比较研究的方法,从以下几个方面较系统地研究了常微分算子理论的发展历程.
一、通过对Sturm和Liouville的工作及其它关于记载这些成果的史料进行分析与研究,从以下几个方面探寻了常微分算子理论的源流：(1)Sturm和Liouville成果的研究背景；(2)分析Sturm和Liouville的工作；(3) Sturm-Liouville理论的意义；(4) Sturm和Liouville工作的后续发展.
二、通过对20世纪早期的一些关于二阶奇异边值问题的文献进行系统分析与考察,从以下几个方面论述了Weyl(1910), Dixon (1912) Stone (1932)和Titchmarsh (1940-1950)的工作对常微分算子理论发展的贡献.我们发现Weyl和Titchmarsh的成果基本上源于经典的实分析和复分析,而Stone的研究工作是Hilbert函数空间抽象理论中自共轭算子与线性常微分方程理论结合的产物.
1.1910年,Weyl不仅开创了奇异S-L微分方程的研究,而且首次考虑了微分方程的分析特征.特别是一些新概念和新成果的提出,使S-L理论在20世纪的发展步入了一个新的发展阶段,也为后来的von Neumann和Stone在微分算子理论方面的研究以及为Titchmarsh应用复变换技巧提供了思想渊源.
2.1912年,Dixon第一次将系数函数p,q,w的连续性条件由Lebesgue可积条件来代替,此Lebesgue可积性条件也是现代微分算子研究中对系数要求最低的条件.
3.1932年,Stone首次在Hilbert函数空问上讨论具有Lebesgue可积系数的二阶微分算子的一般理论.
4. Titchmarsh应用单个复变量函数的展开理论研究了正则情形和奇异情形的S-L边值问题.
三、通过分析与研究关于常微分算子自伴域描述的已有成果,系统地总结了常型和奇异常微分算子自伴域描述的发展脉络.
1.高阶常型微分算子自伴域的描述问题于20世纪50年代彻底解决,1954年Coddington利用矩阵理论和共轭边条件的有关结论,给出了以边条件形式表示的自伴域,这是一个直接的描述结果；同年,Naimark给出了拟微分算子自伴域的描述；1962年,Everitt用微分方程的线性独立解来描述算子的自伴域,在系数足够光滑的条件下,这三个结论是等价的.
2.通过分析奇异微分算子自伴域描述的一些重要成果,比如,Weyl-Titchmarsh自伴域,Everitt自伴域,曹之江-自伴域和孙炯-自伴域,论述了曹之江-自伴域的重要性,它是一种直接而完全的自伴域描述,使得奇异微分算子自伴域描述的问题彻底解决.
四、通过分析和考察大量的关于谱分析方面的文章,主要以离散谱和本质谱的判别为核心梳理了实自伴微分算子,加权的奇异微分算子及J-自伴微分算子离散谱的判别工作和几类特定微分算子本质谱的判别结果.
五、通过挖掘和考察大量的关于亏指数方面的第一手文献,系统地论述了奇异实对称微分算子和复对称微分算子在二阶和高阶情形下极限点型和圆型的判别工作
The theory of ordinary differential operators is playing an important role in the modern quantum mechanic, and is also an important facility in mathematical-physical equations and other applied technology fields. It is a systematic and comprehensive mathematical branch, which includes theories and methods in ordinary differential equations, functional analysis, operator algebra and function spaces. Some principal problems, including description of self-adjoint domains, analysis of spectrum, deficiency index, and inverse spectrum problemetc, are investigated in the theory of ordinary differential operators.
Based on the intensive reading of original sources and research literatures on the subject in consideration, this dissertation investigates systematically and thoroughly into the development history of the theory of ordinary differential operators by using the method of source analysis and the comparative approach of the literature. The main results of this paper are as follows:
First, Based on the brief introduction of the well-known Sturm and Liouville's results, the author systematically and thoroughly discuss the origins of the ordinary differential operators theory. The main contents of this part are as follows:(1) the origins of the Sturm-Liouville theory; (2) the work of Sturm and the work of Liouville; (3) the influence of the Sturm-Liouville theory;(4) the later development of the Sturm-Liouville theory.
Second, By reviewing carefully the early work about the singular Sturm-Liouville boundary value problems, which took place during the years from 1900 to 1950, in this chapterⅢthere are detailed discussions of contributions to from:Weyl(1910), Dixon(1912), Stone (1932) and Titchmarsh (1940-1950). The results of Weyl and Titchmarsh are essentially derived within classical, real and complex mathematical analysis. The results of Stone apply to examples of self-adjoint operators in the abstract theory of Hilbert spaces and in the theory of ordinary linear differential equations. The main contents of this part are as follow:
1. In 1910, the results of Weyl are the first to consider the singular case of the S-L differential equations it is the first structured consideration of the analytical properties of the equation. The rang of new definitions and results is remarkable and set the stage for the full development of S-L theory in the 20th century, as to be seen in the later theory of differential operators in the work of von Neumann and Stone, and in the application of complex variable techniques by Titchmarsh.
2. In 1912, the paper of Dixon seems to be the first paper in which the continuity conditions on the coefficientsp,q,ware replaced by the Lebesgue integrability conditions; these latter conditions are the minimal conditions to be satisfied by p,q,w within the environment given by the Lebesgue integral.
3. In 1932, the results of Stone seems to be the first extended account of the properties of S-L differential operators in the Hilbert function spaces, under the Lebesgue minimal conditions on the coefficients of differential equation.
4. Titchmarsh considered both the regular and singular case of S-L problems by applying the extensive theory of functions of a single complex variable.
Third, By analying the literatures in the past on characterization of self-adjoint domains, the author discusses the important results in the development of the theory of self-adjoint domains. The main content of this part is as follow:
1. In the regular case, in 1954, Coddington used the matrix theory and the conjugate boundary conditions, and given the description of self-adjoint domains of even order symmetric expressions. Naimark considered a symmetric quasi-differential expressions with two regular the boundary conditions. When coefficients are sufficiently smooth, Coddington's condition and Naimark's condition were proved equivalant. In 1962, Everitt given a result using linearly independent solutions of differential equation. This result is similar to Coddington's, and above conditions and Everitt's are equivalent.
2. In the singular case, by studying deeply into the Titchmarsh's result Everitt's result, cao zhi Jiang's result and sun jiong's result, it is pointed out that cao zhi Jiang's result is a complete and direct description of self-adjoint domains with the ones exposed by Everitt as special.
Fourth, based on the analysis of a number of papers on spectra analysis of ordinary differential operators, in particular, the author generalize the important criteria of the discrete spectrum of the real self-adjoint differential operators, weighted differential operators and J- self-adjoint differential operators, and of the essential spectrum of the classes of differential operators.
Fifth, through deep study on the first-hand information, the author systematically and thoroughly discuss the development on the deficiency indices theory for singular symmetric differential operators. In particular two important criterias were drawn as the core of the deficiency indices theory: the limit-point case and the limit-circle case of 2-nd order or higher order differential operators.
引文
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①原著由德文发表,一直未见全文的英文翻译.
②极限点型与极限圆型的属性只与所属算式或算子的系数有关,与其边界条件及谱参数无关.因此,后面出现的算式或算子的极限点型或极限圆型的说法应该是相同的,没有本质上的不同.
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