微分算子乘积的自伴边值问题与一类微分算子的谱特征
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摘要
微分算子自伴边值问题及谱理论是算子理论的重要而基本问题,它是同微分方程、数学物理和量子力学的某些重要问题相联系而发展起来的。本论文研究了微分算子乘积的自伴边值问题及一类微分算子的谱特征。
第一章 简要概述了微分算子研究的背景、进展与本文所研究的问题、使用的方法和获得的若干结果。第二章 引入本文所需要的基本知识、符号和相关的引理。
第三章 讨论了m个由同一n阶对称微分算式生成的赋予某种边界条件的微分算子乘积自伴边值问题,结合常微分算子自伴扩张的一般构造理论,分别给出了两个四阶微分算子、两个n阶微分算子、m个n阶微分算子乘积自伴边条件的解析刻划,得到了乘积微分算子是自伴的充分必要条件及与乘积算子自伴性有关的一些有益结果。第四章对区间(a,b)(-∞≤a 第五章 运用算子方法,研究了复系数J-对称微分算式生成的J-自伴微分算子谱的离散性,分别得到了一类J-自伴微分算子谱离散的充分条件与必要条件,为判断微分算子谱的离散性提供了若干准则。同时,在加权空间中,讨论了由复周期系数J-对称微分算式生成的J-自伴微分算子的本质谱,给出了一类J-自伴微分算子本质谱的存在区域。
最后,在第六章运用矩阵微分方程理论,考察了二维向量自伴Sturm-Liouville微分方程的谱,证明了在某些假设条件下,这类微分方程仅有有限多个二重特征值,进一步估计出一个下界mQ,使得当n>mQ时,特征值λ_n都是简单(一重)的。作为结果的应用,得到了两个数量型自伴Sturm-Liouville微分方程具有有限多个共同特征值的一个充分条件,并对此共同特征值的个数给出了估计。
The self-adjoint boundary-value problems and spectral theory of differential operators are important and fundamental problems in the operator theory. The self-adj oint boundary-value problems for products of differential operators and spectral properties of a class of differential operators are investigated in this dissertation.Chapter 1 deals with the background and advance of the study on the differential operators, and the main topic, some important results obtained and employed methods in this paper. Chapter 2 is some preliminaries, symbols and some lemmas.In Chapter 3, we discuss the self-adjoint boundary-value problems for products of m differential operators generated by the same symmetric differential expression of order n defined on [a, b) (-∞ < a < b ≤ ∞), endowed with some boundary conditions. Combining the general construction theory of self-adjoint extensions of ordinary differential operators, we give, respectively, analytic characterization for self-adjoint boundary conditions of products of two 4th-order differential operators, two nth-order differential operators, and m nth-order differential operators at the regular (b < ∞) and the singular (b = ∞) cases, and obtain the necessary and sufficient conditions for self-adjointness of products of differential operators. Furthermore, some useful results concerning self-adjointness of the product operator are obtained. Meanwhile, in Chapter 4, for a regular symmetric vector-valued differential expression l(y) of order n on (a,b) (-∞ < a < b ≤ ∞), under the assumption that the power expression l~2(y) is partially separated in weighted function space L_r~2(a,b), the boundary conditions determining Friedrichs extension of the minimal operator generated by l~2(y) are identified explicitly.In Chapter 5, using the approach of operators, we discuss the discreteness of the spectrum of J—self-adjoint differential operators that are generated by J—symmetric differential expression with complex-valued coefficients. Some criteria for the discrete spectrum of J—self-adjoint differential operators are obtained, which provide some criteria for the discreteness of spectrum of the differential operators. Meanwhile, we also investigate the essential spectrum of J—self-adjoint differential operators that are extended by J—symmetric differential expression with complex-valued periodic function coefficients where the underlying Hilbert space is weighted. The existence region for the essential spectrum of J—self-adjoint differential operators defined above is located.
第一章 简要概述了微分算子研究的背景、进展与本文所研究的问题、使用的方法和获得的若干结果。第二章 引入本文所需要的基本知识、符号和相关的引理。
第三章 讨论了m个由同一n阶对称微分算式生成的赋予某种边界条件的微分算子乘积自伴边值问题,结合常微分算子自伴扩张的一般构造理论,分别给出了两个四阶微分算子、两个n阶微分算子、m个n阶微分算子乘积自伴边条件的解析刻划,得到了乘积微分算子是自伴的充分必要条件及与乘积算子自伴性有关的一些有益结果。第四章对区间(a,b)(-∞≤a
最后,在第六章运用矩阵微分方程理论,考察了二维向量自伴Sturm-Liouville微分方程的谱,证明了在某些假设条件下,这类微分方程仅有有限多个二重特征值,进一步估计出一个下界mQ,使得当n>mQ时,特征值λ_n都是简单(一重)的。作为结果的应用,得到了两个数量型自伴Sturm-Liouville微分方程具有有限多个共同特征值的一个充分条件,并对此共同特征值的个数给出了估计。
The self-adjoint boundary-value problems and spectral theory of differential operators are important and fundamental problems in the operator theory. The self-adj oint boundary-value problems for products of differential operators and spectral properties of a class of differential operators are investigated in this dissertation.Chapter 1 deals with the background and advance of the study on the differential operators, and the main topic, some important results obtained and employed methods in this paper. Chapter 2 is some preliminaries, symbols and some lemmas.In Chapter 3, we discuss the self-adjoint boundary-value problems for products of m differential operators generated by the same symmetric differential expression of order n defined on [a, b) (-∞ < a < b ≤ ∞), endowed with some boundary conditions. Combining the general construction theory of self-adjoint extensions of ordinary differential operators, we give, respectively, analytic characterization for self-adjoint boundary conditions of products of two 4th-order differential operators, two nth-order differential operators, and m nth-order differential operators at the regular (b < ∞) and the singular (b = ∞) cases, and obtain the necessary and sufficient conditions for self-adjointness of products of differential operators. Furthermore, some useful results concerning self-adjointness of the product operator are obtained. Meanwhile, in Chapter 4, for a regular symmetric vector-valued differential expression l(y) of order n on (a,b) (-∞ < a < b ≤ ∞), under the assumption that the power expression l~2(y) is partially separated in weighted function space L_r~2(a,b), the boundary conditions determining Friedrichs extension of the minimal operator generated by l~2(y) are identified explicitly.In Chapter 5, using the approach of operators, we discuss the discreteness of the spectrum of J—self-adjoint differential operators that are generated by J—symmetric differential expression with complex-valued coefficients. Some criteria for the discrete spectrum of J—self-adjoint differential operators are obtained, which provide some criteria for the discreteness of spectrum of the differential operators. Meanwhile, we also investigate the essential spectrum of J—self-adjoint differential operators that are extended by J—symmetric differential expression with complex-valued periodic function coefficients where the underlying Hilbert space is weighted. The existence region for the essential spectrum of J—self-adjoint differential operators defined above is located.
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