无穷度量图上Sturm-Liouville算子的谱性质

发布时间：2018-06-21 11:37

本文选题：量子图 + 正则度量树　；参考：《天津大学》2016年博士论文

【摘要】：度量图上的微分算子是研究介观物理与化学结构问题的抽象数学模型,在化学、粒子物理及纳米技术等学科中应用十分广泛.经过近几十年的发展,度量图上微分算子理论已经成为微分方程理论以及微分算子谱理论的重要组成部分.本文主要研究无穷度量图上Sturm-Liouville算子的自伴性以及无穷正则度量树上Sturm-Liouville算子的谱性质,全文分为六个部分,内容如下:第一章为绪论,介绍了度量图上的微分算子的研究背景、研究现状以及本文的主要工作.第二章介绍了本文所涉及的基本概念以及相关定理.第三章主要研究了三类局部顶点条件:系数矩阵秩为δ(v)的顶点条件、自伴顶点条件和J-自伴顶点条件.给出了这三类顶点条件的性质并且讨论了它们分别确定的定义域所构成空间的几何结构.当微分形式对称时,给出了紧致度量图上Sturm-Liouville算子的自伴条件和非紧致度量图上Sturm-Liouville算子的Glazman-Povzner-Wienholtz型自伴条件.当微分形式J-对称时,详细描述了度量图上局部Sturm-Liouville算子的J-伴随算子及J-自伴扩张.第四章主要研究了无穷正则度量树上带有δ-型条件的Schr?dinger算子的自伴性及其谱性质.首先,给出了该算子的本质自伴判定条件,且证明了该算子酉等价于一列带有转移条件的辅助算子的直和.其次,构造了带有转移条件的辅助算子所对应的二次型,给出了辅助算子的Molchanov谱离散判定准则.基于树上Schr?dinger算子与其辅助算子谱之间的关系,得到了树上带有δ-型条件的Schr?dinger算子谱纯离散的充分必要条件.最后,根据二次型扰动的紧性,得到了树上带有δ-型条件的Schr?dinger算子本质谱稳定的判定条件和负谱下半有界且离散的判定条件.第五章主要考虑了无穷正则度量树上带有δ'-型条件的Schr?dinger算子.因为该算子的定义域包含于直和空间(?),并不包含于树上Sobolev空间(?),所以构造了一列带有转移条件的辅助算子及其所对应的二次型,并证明了一系列嵌入不等式,进而通过嵌入算子的紧性研究了树上Schr?dinger算子谱的纯离散性.所得结果将Molchanov离散谱判定准则推广至具有转移条件的Schr?dinger算子.基于树上带有δ'-型条件的Schr?dinger算子与辅助算子谱之间的关系得到了树上带有δ'-型条件的Schr?dinger算子谱纯离散的充分必要条件和本质谱稳定的判定条件.第六章讨论了边长下确界为0的无穷度量树上Sturm-Liouville算子的自伴性及其谱性质.本章证明了树上Sturm-Liouville算子酉等价于一列带有转移条件的辅助算子的直和,借助于这一列辅助算子,证明了树上算子的自伴性.然后利用无穷区间覆盖定理,证明了加权函数空间上的不等式,得到了带有转移条件的Sturm-Liouville算子谱离散的充分必要条件,进而得到了无穷度量树上Sturm-Liouville算子的Molchanov离散准则。
[Abstract]:Differential operators on metric graphs are abstract mathematical models for studying mesoscopic physics and chemical structures. They are widely used in chemistry, particle physics and nanotechnology. With the development of recent decades, differential operator theory on metric graph has become an important part of differential equation theory and differential operator spectrum theory. In this paper, we study the self-adjoint of Sturm-Liouville operators on infinite metric graphs and the spectral properties of Sturm-Liouville operators on infinite metric trees. The whole paper is divided into six parts. The contents are as follows: the first chapter is an introduction, and introduces the research background of differential operators on metric graphs. The present situation of the research and the main work of this paper. The second chapter introduces the basic concepts and related theorems involved in this paper. In chapter 3, we study three kinds of local vertex conditions: the rank of coefficient matrix is 未 v), the self-adjoint vertex condition and the J-self-adjoint vertex condition. The properties of these three kinds of vertex conditions are given and the geometric structure of the space formed by the defined domain determined by them is discussed. The self-adjoint conditions for Sturm-Liouville operators on compact metric graphs and Glazman-Povzner-Wienholtz type self-adjoint conditions for Sturm-Liouville operators on non-compact metric graphs are given when differential forms are symmetric. When the differential form is J-symmetric, the J-adjoint operators and J-self-adjoint extensions of local Sturm-Liouville operators on metric graphs are described in detail. In chapter 4, we study the self-adjoint and spectral properties of Schrndinger operators with 未 -type conditions on infinite regular metric trees. Firstly, the essential self-adjoint condition of the operator is given, and it is proved that the operator is unitary equivalent to the direct sum of a series of auxiliary operators with transfer conditions. Secondly, the quadratic form of auxiliary operator with transfer condition is constructed, and the Molchanov spectrum discretization criterion of auxiliary operator is given. Based on the relationship between the Schrndinger operator and its auxiliary operator spectrum on a tree, a sufficient and necessary condition for the pure discretization of the Schrrdinger operator spectrum on a tree with 未 -type condition is obtained. Finally, according to the compactness of the quadratic perturbation, the conditions for the stability of the intrinsic mass spectrum of the Schrdinger operator with 未 -type condition on the tree and the semi-bounded and discrete conditions for the negative spectrum are obtained. In chapter 5, we consider Schrndinger operators with 未 -type conditions on infinite regular metric trees. Because the domain of the operator is contained in the direct sum space and not in the Sobolev space on the tree, a series of auxiliary operators with transfer conditions and their corresponding quadratic forms are constructed, and a series of embedding inequalities are proved. Furthermore, the pure discreteness of the spectrum of Schrndinger operators on the tree is studied by the compactness of embedding operators. The result extends the Molchanov discrete spectrum criterion to Schrndinger operator with transfer condition. Based on the relationship between Schrndinger operator with 未 -type condition and the spectrum of auxiliary operator on a tree, a sufficient and necessary condition for pure discretization of Schrndinger operator spectrum on a tree with 未 -type condition and a criterion for the stability of intrinsic mass spectrum are obtained. In chapter 6, we discuss the self-adjoint property and spectral properties of Sturm-Liouville operators on infinite metric trees with the lower bound of the edge length being 0. In this chapter, we prove that the Sturm-Liouville operator on a tree is unitary equivalent to the direct sum of a sequence of auxiliary operators with transfer conditions. With the aid of this set of auxiliary operators, we prove the self-adjoint property of the tree operators. Then, by using the infinite interval covering theorem, we prove the inequalities on the weighted function space, obtain the necessary and sufficient conditions for the spectral discretization of Sturm-Liouville operators with transfer conditions, and then obtain the Molchanov discretization criteria for Sturm-Liouville operators on infinite metric trees.
【学位授予单位】：天津大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.3

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