一类内部点条件含有谱参数的二阶微分算子的特征值

数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 815-828.

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一类内部点条件含有谱参数的二阶微分算子的特征值

刘薇(),许美珍^*()

内蒙古工业大学理学院呼和浩特 010051

收稿日期:2023-05-24 修回日期:2024-01-25 出版日期:2024-08-26 发布日期:2024-07-26
通讯作者: *许美珍, E-mail:xumeizhen1969@163.com
作者简介:刘薇, E-mail:17743779459@163.com
基金资助:
国家自然科学基金(12261066);内蒙古自然科学基金(2021MS01020);内蒙古自然科学基金(2023LHMS01015);内蒙古自治区直属高校基本科研业务费(JY20240043)

Eigenvalues of a Class of Second-Order Differential Operator with Eigenparameters Dependent Internal Point Conditions

Liu Wei(),Xu Meizhen^*()

College of Sciences, Inner Mongolia University of Technology, Hohhot 010051

Received:2023-05-24 Revised:2024-01-25 Online:2024-08-26 Published:2024-07-26
Supported by:
NSFC(12261066);NSF of Inner Mongolia(2021MS01020);NSF of Inner Mongolia(2023LHMS01015);Basic Science Research Fund of the Universities Directly Under the lnner Mongolia Autonomous Region(JY20240043)

摘要/Abstract

摘要：

该文主要讨论了一类内部点条件含有谱参数的二阶微分算子的自伴性和特征值的依赖性. 首先, 在适当的 Hilbert 空间中定义一个与问题相关的线性算子 $T$, 将所要研究的问题转化为对此空间中算子 $T$ 的研究, 并根据自伴算子的定义证明了算子 $T$ 是自伴的. 另外, 在自伴的基础上, 证明了特征值不仅连续依赖而且可微依赖于问题的各个参数, 并给出相应的微分表达式. 同时, 还讨论了特征值关于问题部分参数的单调性.

关键词: 内部点条件, 谱参数, 自伴性, 特征值的依赖性, 特征值的单调性

Abstract:

This paper mainly discusses the self-adjointness and eigenvalue dependence of a class of second-order differential operator with internal point conditions containing an eigenparameter. First, a problem-related linear operator $T$ is defined in an appropriate Hilbert space, and the study of the problem to be transformed into the research of the operator $T$ in this space, and the operator $T$ is proved to be self-adjoint according to the definition of self-adjoint operator. In addition, on the basis of self-adjoint, it is proved that the eigenvalues are not only continuously dependent but also differentiable on each parameter of the problem, and the corresponding differential expressions are given. Meanwhile, the monotonicity of the eigenvalues with respect to the part parameters of the problem is also discussed.

Key words: Internal point conditions, Eigenparameters, Self-adjointness, Dependence of eigenvalue, Monotonicity of eigenvalues

中图分类号:

O175.3

刘薇, 许美珍. 一类内部点条件含有谱参数的二阶微分算子的特征值[J]. 数学物理学报, 2024, 44(4): 815-828.

Liu Wei, Xu Meizhen. Eigenvalues of a Class of Second-Order Differential Operator with Eigenparameters Dependent Internal Point Conditions[J]. Acta mathematica scientia,Series A, 2024, 44(4): 815-828.

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一类内部点条件含有谱参数的二阶微分算子的特征值

Eigenvalues of a Class of Second-Order Differential Operator with Eigenparameters Dependent Internal Point Conditions

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[2]	闫文文,许美珍. 边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性[J]. 数学物理学报, 2022, 42(3): 671-693.