无穷维Hamilton算子的谱与特征函数系的完备性
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摘要
本学位论文以无穷维Hamilton算子特征函数系(辛正交系)的完备性为主题,围绕着无穷维Hamilton算子的谱理论以及完备不定度规空间中极大确定不变子空间的存在性问题开展研究工作,从而拓广了Strum-Liouville问题以及按特征函数展开的求解方法,为Hamilton体系下采用分离变量法提供了理论保障。
无穷维Hamilton算子特征函数系(辛正交系)的完备性问题是无穷维Hamilton算子理论以及无穷维Hamilton系统中的重要问题。对于分离变量以后可导向Strum-Liouville问题的偏微分方程,分离变量法是一种十分有效的求解方法。但是,无穷维Hamilton算子一般情况下是非自伴算子,因此在Hamilton体系下的分离变量法是否适合和正确的问题显得格外重要。然而以上问题的理论基础是无穷维Hamilton算子特征函数系(辛正交系)的完备性问题。因此,本文充分利用无穷维Hamilton算子特征函数系的辛正交性以及一类无穷维Hamilton算子的特征值正负成对出现的独特性质,给出了一类无穷维Hamilton算子特征函数系(辛正交系)的Cauchy主值意义下完备的充分条件,在此基础上,对这类无穷维Hamilton正则系统采用Cauchy主值意义下的分离变量法(即,分离变量法采用Cauchy主值意义下的叠加原理)得到了Cauchy主值意义下完备的解。这一工作对于解决无穷维Hamilton正则系统的求解乃至一般的偏微分方程的求解问题提供了新方法、新思想,具有极高的理论价值与实际意义。
要解决更一般的无穷维Hamilton正则系统的求解问题,须考虑它所对应的一般无穷维Hami lton算子的特性,这个问题属于线性算子理论范畴。我们知道,线性算子的谱分析是泛函分析的重要组成部分,是线性算子理论的灵魂,它的中心课题是谱分解理论。因此,本文中把无穷维Hamilton算子的谱理论放在了非常重要的位置,给出了上三角型无穷维Hamilton算子的谱集以及连续谱只和主对元有关的充要条件,从而为彻底解决上三角型无穷维Hamilton算子的谱补问题和谱扰动问题提供了必要的准备;为了解决无穷维Hamilton算子生成强连续半群的问题,又给出了无穷维Hamilton算子只有纯续谱的充分条件。除此之外,当系统导出的算子可逆时,对半解析法提供了强有力的保障,此时,偏微分方程可化成常微分方程,因此无穷维Hamilton算子的可逆性问题也显得很重要,而这个问题的本质是零点是否包含于正则点集的问题。从而,本文利用非负Hamilton算子的结构特性,运用内部项刻画了一般的非负Hamilton算子的可逆性问题,解决了非负Hamilton算子何时具有紧域解式的问题。值得注意的是,在刻画谱集的分布范围时数值域有着非常重要的应用,因为有界线性算子的数值域闭包包含谱集,然而,最近发现,对有界线性算子来说二次数值域不仅是数值域的子集而且它的闭包也包含谱集,因此,刻画谱集时二次数值域能提供比数值域更好的信息。基于以上原因,本文又研究了一类无界无穷维Hamilton算子的数值域和二次数值域,并给出了不仅数值域的闭包包含谱集,而且二次数值域的闭包也包含谱集的结论。
本文还研究了完备不定度规空间中无穷维Hamilton算子的谱理论.不定度规空间上的算子理论并不是Hilbert空间上算子理论逻辑上的推广,而是有着深厚的基础的。它的应用涉及到物理学、数学及力学方面。由于无穷维Hamilton算子的特殊性,引进适当的不定度规以后无穷维Hamilton算子会变成不定度规意义下反自伴算子,而此时,它的性质与完备不定度规空间中自伴算子的性质非常接近,因此可以得到许多有意义的结论。在此基础上,本文又给出了无穷维Hamilton算子在完备不定度规空间中存在极大确定不变子空间的充分条件。
其次,自从H.Weyl在1909年发现有界自伴算子的孤立的有限重特征值集合与Weyl谱在谱集中的补集重叠以后(即,著名的Weyl型定理),J.Schwartz,S.Berberian等许多学者研究哪些算子满足Weyl型定理,于是满足Weyl型定理的算子范围不断扩大。但是,大部分成果均以有界算子为研究对象,关于无界算子的Weyl型定理的结论非常少见。因此本文给出了具有扰动的无界自伴线性算子满足Weyl型定理的充分条件,仅而得到了紧算子满足Weyl型定理的充分条件。
全文分为七章,第一章介绍了选题意义和我们的主要工作;第二章给出了上三角型无穷维Hamilton算子的谱的性质,并讨论了无穷维Hamilton算子特征值问题;第三章是无穷维Hamilton算子特征函数系的Cauchy主值意义下完备问题;第四章是非负Hamilton算子的可逆性问题;第五章研究了一类无穷维Hamilton算子的数值域及二次数值域的性质;第六章研究了具有扰动的无界自伴线性算子何时满足Weyl型定理的问题;第七章是完备不定度规空间中无穷维Hamilton算子的谱理论以及极大确定不变子空间存在性问题。
This dissertation focuses on the completeness of the eigenfunctions systems(symplectic orthogonal system) of infinite dimensional Hamiltonian operators and researches into the spectral theory and on existence of maximal definite invariant subspace in Krein space,which developes the Strum-Liouville problems and the methods of eigcnfunctions expansion and provides a theoretical basis for employing the method of separation of variables based on Hamiltonian systems.
In theories of infinite dimensional Hamiltonian operators and infinite dimensional Hamiltonian systelns,completeness of the eigenfunctions systems(symplectic orthogohal system) of the infinite dimensional Hamiltonian operators is very important problem. The traditional method of separation of variable is effective to solve partial differential equations which can be transformed into the Strum-Liouville problem after separating variables.However,infinite dimensional Hamiltonian operator is non-selfadjoint operator in generally,therefore,to employ the method of separation of variables based on Hamiltonian systems,the completeness of the eigenfunctions systems(symplectic orthogonal system) of the infinite dimensional Hamiltonian operators must be solved. Thus,we obtain the sufficient conditions of the completeness in the sense of Cauchy Principal Value of the eigenfunctions systems of the infinite dimensional Hamiltonian operator by taking advatage of the symplectic orthogonality of eigenfunctions systems and the property of existing real eigenvalues or pure imaginary eigenvalues only and appear pairwise according to positive and negative,consequently,we get solutions of complete in sense of Cauchy principal value.This works give a new method and new idea to solve infinite dimensional Hamiltonian system and even ordinary partial differential equations and possess high theoretical values and practical significance.
To solve more general infinite dimensional Hamiltonian systems,we must study the properties of general infinite dimensional Hamiltonian operator,which belongs to areas of linear operator theory.As far as we know,spectral analysis of linear operator is important component of functional analysis and soul of linear operator theory,its centre subject is spectral decomposition.Therefore,in this paper,we also focus on spectrum of infinite dimensional Hamiltonian operator and obtain spectral properties of upper triangular infinite dimensional Hamiltonian operator,which provides necessary preparations for solving completion problem and spectral perturbation problem of upper triangular infinite dimensional Hamiltonian operator;To solve the problem of infinite dimensional Hamiltonian operator generates C_0 Semi-group,we also obtain the sufficient conditions of infinite dimensional Hamiltonian operator exists pure imaginary spectrum only;In addition,when the operator is invertible,it provide theoretical foundations for semi-analytical method and the partial differential equations can be transformed into ordinary differential equations,therefore,the problem of invertibility of infinite dimensional Hamiltonian operator is very important and the nature of problem is zero point whether belongs to regular set.Thereby,taking full advantage of structure of non-negative Hamiltonian operator,the sufficient conditions for nonnegative Hamiltonian operators exist everywhere defined bounded inverse are given. It is worth noting that the notion of numerical range is important in various applications, since it was used to as a tool in order to localize the spectrum of operators,that is to say,the closure of numerical range contains the spectral set.However.recently H.Langer found that the quadratic numerical range of bounded operator is a subset of the numerical range and that its closure still contains the spectral set.Thus,in general, it gives better information about the location of the spectrum of bounded linear operator than the numerical range.So,in this paper we study the quadratic numerical range and numerical range of a class of unbounded infinite dimensional Hamiltonian operators and the conclusion that not only the closure of the numerical range contains the spectral set but also the closure of the quadratic numerical range contains the spectral set is shown.
We also investigate the spectral theory of infinite dimensional Hamiltonian operators in complete indefinite metric space.Linear operator theory in indefinite metric space is not a logical promotion of linear operator theory in Hilbert space,but has profound theoretical basis,its application including physics,mathematics and mechanics. In view of particularities of infinite dimensional Hamiltonian operators,after introducing appropriate indefinite metric,it can become anti-selfadjoint operator;therefore, we can draw meaningful conclusions.Furthermore,the sufficient conditions of infinite dimensional Hamiltonian operator exists maximal definite invariant subspace is given.
In 1909,H.Weyl discovered that complement in the spectrum of the Weyl spectrum of bounded selfadjoint linear operator coincides with the isolated eigenvalue of finite multiplicity.Today this result is known as Weyl's theorem and it has been studied by numerous authors,such as J.Schwartz,S.Berberian,and extended from bounded selfadjoint operator to other class of bounded operator.But most of results are Weyl's theorem for bounded operators and about unbounded operators are very rare.Hence in this paper we consider how Weyl's theorem survives for unbounded self-adjoint operator under small perturbations and the sufficient conditions of compact operator survives Weyl's theorem are given.
This paper contains seven chapters.In first chapter,we introduce the significance of topics and main results we obtained;In second chapter the spectral properties of upper triangular Hamiltonian operators are given and the eigenvalue problems of infinite dimensional Hamiltonian operators are discussed;In third chapter,completeness in the sense of Cauchy principal value of the eigenfunctions systems(symplectic orthogonal system) of the infinite dimensional Halniltonian operators is studied;In fourth chapter we investigate the invertibility of non-negative Hamiltonian operators; In chapter fifth,the properties of numerical range and quadratic numerical range of infinite dimensional Hamiltonian operators are considered;In sixth chapter,the Weyl's theorem for unbounded operators under small perturbations is studied;In last chapter we introduce spectral theory of infinite dimensional Hamiltonian operators in Krein space.
无穷维Hamilton算子特征函数系(辛正交系)的完备性问题是无穷维Hamilton算子理论以及无穷维Hamilton系统中的重要问题。对于分离变量以后可导向Strum-Liouville问题的偏微分方程,分离变量法是一种十分有效的求解方法。但是,无穷维Hamilton算子一般情况下是非自伴算子,因此在Hamilton体系下的分离变量法是否适合和正确的问题显得格外重要。然而以上问题的理论基础是无穷维Hamilton算子特征函数系(辛正交系)的完备性问题。因此,本文充分利用无穷维Hamilton算子特征函数系的辛正交性以及一类无穷维Hamilton算子的特征值正负成对出现的独特性质,给出了一类无穷维Hamilton算子特征函数系(辛正交系)的Cauchy主值意义下完备的充分条件,在此基础上,对这类无穷维Hamilton正则系统采用Cauchy主值意义下的分离变量法(即,分离变量法采用Cauchy主值意义下的叠加原理)得到了Cauchy主值意义下完备的解。这一工作对于解决无穷维Hamilton正则系统的求解乃至一般的偏微分方程的求解问题提供了新方法、新思想,具有极高的理论价值与实际意义。
要解决更一般的无穷维Hamilton正则系统的求解问题,须考虑它所对应的一般无穷维Hami lton算子的特性,这个问题属于线性算子理论范畴。我们知道,线性算子的谱分析是泛函分析的重要组成部分,是线性算子理论的灵魂,它的中心课题是谱分解理论。因此,本文中把无穷维Hamilton算子的谱理论放在了非常重要的位置,给出了上三角型无穷维Hamilton算子的谱集以及连续谱只和主对元有关的充要条件,从而为彻底解决上三角型无穷维Hamilton算子的谱补问题和谱扰动问题提供了必要的准备;为了解决无穷维Hamilton算子生成强连续半群的问题,又给出了无穷维Hamilton算子只有纯续谱的充分条件。除此之外,当系统导出的算子可逆时,对半解析法提供了强有力的保障,此时,偏微分方程可化成常微分方程,因此无穷维Hamilton算子的可逆性问题也显得很重要,而这个问题的本质是零点是否包含于正则点集的问题。从而,本文利用非负Hamilton算子的结构特性,运用内部项刻画了一般的非负Hamilton算子的可逆性问题,解决了非负Hamilton算子何时具有紧域解式的问题。值得注意的是,在刻画谱集的分布范围时数值域有着非常重要的应用,因为有界线性算子的数值域闭包包含谱集,然而,最近发现,对有界线性算子来说二次数值域不仅是数值域的子集而且它的闭包也包含谱集,因此,刻画谱集时二次数值域能提供比数值域更好的信息。基于以上原因,本文又研究了一类无界无穷维Hamilton算子的数值域和二次数值域,并给出了不仅数值域的闭包包含谱集,而且二次数值域的闭包也包含谱集的结论。
本文还研究了完备不定度规空间中无穷维Hamilton算子的谱理论.不定度规空间上的算子理论并不是Hilbert空间上算子理论逻辑上的推广,而是有着深厚的基础的。它的应用涉及到物理学、数学及力学方面。由于无穷维Hamilton算子的特殊性,引进适当的不定度规以后无穷维Hamilton算子会变成不定度规意义下反自伴算子,而此时,它的性质与完备不定度规空间中自伴算子的性质非常接近,因此可以得到许多有意义的结论。在此基础上,本文又给出了无穷维Hamilton算子在完备不定度规空间中存在极大确定不变子空间的充分条件。
其次,自从H.Weyl在1909年发现有界自伴算子的孤立的有限重特征值集合与Weyl谱在谱集中的补集重叠以后(即,著名的Weyl型定理),J.Schwartz,S.Berberian等许多学者研究哪些算子满足Weyl型定理,于是满足Weyl型定理的算子范围不断扩大。但是,大部分成果均以有界算子为研究对象,关于无界算子的Weyl型定理的结论非常少见。因此本文给出了具有扰动的无界自伴线性算子满足Weyl型定理的充分条件,仅而得到了紧算子满足Weyl型定理的充分条件。
全文分为七章,第一章介绍了选题意义和我们的主要工作;第二章给出了上三角型无穷维Hamilton算子的谱的性质,并讨论了无穷维Hamilton算子特征值问题;第三章是无穷维Hamilton算子特征函数系的Cauchy主值意义下完备问题;第四章是非负Hamilton算子的可逆性问题;第五章研究了一类无穷维Hamilton算子的数值域及二次数值域的性质;第六章研究了具有扰动的无界自伴线性算子何时满足Weyl型定理的问题;第七章是完备不定度规空间中无穷维Hamilton算子的谱理论以及极大确定不变子空间存在性问题。
This dissertation focuses on the completeness of the eigenfunctions systems(symplectic orthogonal system) of infinite dimensional Hamiltonian operators and researches into the spectral theory and on existence of maximal definite invariant subspace in Krein space,which developes the Strum-Liouville problems and the methods of eigcnfunctions expansion and provides a theoretical basis for employing the method of separation of variables based on Hamiltonian systems.
In theories of infinite dimensional Hamiltonian operators and infinite dimensional Hamiltonian systelns,completeness of the eigenfunctions systems(symplectic orthogohal system) of the infinite dimensional Hamiltonian operators is very important problem. The traditional method of separation of variable is effective to solve partial differential equations which can be transformed into the Strum-Liouville problem after separating variables.However,infinite dimensional Hamiltonian operator is non-selfadjoint operator in generally,therefore,to employ the method of separation of variables based on Hamiltonian systems,the completeness of the eigenfunctions systems(symplectic orthogonal system) of the infinite dimensional Hamiltonian operators must be solved. Thus,we obtain the sufficient conditions of the completeness in the sense of Cauchy Principal Value of the eigenfunctions systems of the infinite dimensional Hamiltonian operator by taking advatage of the symplectic orthogonality of eigenfunctions systems and the property of existing real eigenvalues or pure imaginary eigenvalues only and appear pairwise according to positive and negative,consequently,we get solutions of complete in sense of Cauchy principal value.This works give a new method and new idea to solve infinite dimensional Hamiltonian system and even ordinary partial differential equations and possess high theoretical values and practical significance.
To solve more general infinite dimensional Hamiltonian systems,we must study the properties of general infinite dimensional Hamiltonian operator,which belongs to areas of linear operator theory.As far as we know,spectral analysis of linear operator is important component of functional analysis and soul of linear operator theory,its centre subject is spectral decomposition.Therefore,in this paper,we also focus on spectrum of infinite dimensional Hamiltonian operator and obtain spectral properties of upper triangular infinite dimensional Hamiltonian operator,which provides necessary preparations for solving completion problem and spectral perturbation problem of upper triangular infinite dimensional Hamiltonian operator;To solve the problem of infinite dimensional Hamiltonian operator generates C_0 Semi-group,we also obtain the sufficient conditions of infinite dimensional Hamiltonian operator exists pure imaginary spectrum only;In addition,when the operator is invertible,it provide theoretical foundations for semi-analytical method and the partial differential equations can be transformed into ordinary differential equations,therefore,the problem of invertibility of infinite dimensional Hamiltonian operator is very important and the nature of problem is zero point whether belongs to regular set.Thereby,taking full advantage of structure of non-negative Hamiltonian operator,the sufficient conditions for nonnegative Hamiltonian operators exist everywhere defined bounded inverse are given. It is worth noting that the notion of numerical range is important in various applications, since it was used to as a tool in order to localize the spectrum of operators,that is to say,the closure of numerical range contains the spectral set.However.recently H.Langer found that the quadratic numerical range of bounded operator is a subset of the numerical range and that its closure still contains the spectral set.Thus,in general, it gives better information about the location of the spectrum of bounded linear operator than the numerical range.So,in this paper we study the quadratic numerical range and numerical range of a class of unbounded infinite dimensional Hamiltonian operators and the conclusion that not only the closure of the numerical range contains the spectral set but also the closure of the quadratic numerical range contains the spectral set is shown.
We also investigate the spectral theory of infinite dimensional Hamiltonian operators in complete indefinite metric space.Linear operator theory in indefinite metric space is not a logical promotion of linear operator theory in Hilbert space,but has profound theoretical basis,its application including physics,mathematics and mechanics. In view of particularities of infinite dimensional Hamiltonian operators,after introducing appropriate indefinite metric,it can become anti-selfadjoint operator;therefore, we can draw meaningful conclusions.Furthermore,the sufficient conditions of infinite dimensional Hamiltonian operator exists maximal definite invariant subspace is given.
In 1909,H.Weyl discovered that complement in the spectrum of the Weyl spectrum of bounded selfadjoint linear operator coincides with the isolated eigenvalue of finite multiplicity.Today this result is known as Weyl's theorem and it has been studied by numerous authors,such as J.Schwartz,S.Berberian,and extended from bounded selfadjoint operator to other class of bounded operator.But most of results are Weyl's theorem for bounded operators and about unbounded operators are very rare.Hence in this paper we consider how Weyl's theorem survives for unbounded self-adjoint operator under small perturbations and the sufficient conditions of compact operator survives Weyl's theorem are given.
This paper contains seven chapters.In first chapter,we introduce the significance of topics and main results we obtained;In second chapter the spectral properties of upper triangular Hamiltonian operators are given and the eigenvalue problems of infinite dimensional Hamiltonian operators are discussed;In third chapter,completeness in the sense of Cauchy principal value of the eigenfunctions systems(symplectic orthogonal system) of the infinite dimensional Halniltonian operators is studied;In fourth chapter we investigate the invertibility of non-negative Hamiltonian operators; In chapter fifth,the properties of numerical range and quadratic numerical range of infinite dimensional Hamiltonian operators are considered;In sixth chapter,the Weyl's theorem for unbounded operators under small perturbations is studied;In last chapter we introduce spectral theory of infinite dimensional Hamiltonian operators in Krein space.
引文
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