向量型 Sturm-Liouville问题的特征值重数及逆结点问题

数学物理学报 ›› 2023, Vol. 43 ›› Issue (3): 669-679.

向量型 Sturm-Liouville问题的特征值重数及逆结点问题

刘肖云¹(),史国良²,闫军^2,^*()

¹安阳工学院数学与信息科学学院河南安阳 455000
²天津大学数学学院天津 300354

收稿日期:2021-08-23 修回日期:2023-02-06 出版日期:2023-06-26 发布日期:2023-06-01
通讯作者: 闫军 E-mail:xyl.hb@163.com;jun.yan@tju.edu.cn
作者简介:刘肖云, E-mail: xyl.hb@163.com
基金资助:
国家自然科学基金(12001153);国家自然科学基金(62065015);河北省自然科学基金(F2022407007);河北省高等学校科学技术研究项目(ZC2023122)

The Multiplicities of Eigenvalues and Inverse Nodal Problem of a Vectorial Sturm-Liouville Problem

Liu Xiaoyun¹(),Shi Guoliang²,Yan Jun^2,^*()

¹School of Mathematics and Information Science, Anyang Institute of Techology, Henan Anyang 455000
²School of Mathematical Sciences, Tianjin University, Tianjin 300354

Received:2021-08-23 Revised:2023-02-06 Online:2023-06-26 Published:2023-06-01
Contact: Jun Yan E-mail:xyl.hb@163.com;jun.yan@tju.edu.cn
Supported by:
National Natural Science Foundation of China(12001153);National Natural Science Foundation of China(62065015);Natural Science Foundation of Hebei Province(F2022407007);Science and Technology Research Project of Colleges and Universities in Hebei Province(ZC2023122)

摘要/Abstract

摘要：

该文研究定义在区间 $(0,1)$ 上具有 Dirichlet 边界条件的 $m$ 维向量型 Sturm-Liouville 问题. 首先, 讨论矩阵值势函数与特征值重数之间的关系,证明如果矩阵 $\int_{0}^{1}Q(x){\rm d}x$ 的特征值重数至多为 $k$ $(1\leq k\leq m-1)$, 那么除有限个特征值外, 向量型问题的特征值重数也至多为 $k$.然后, 采用一个不同的思路研究逆结点问题, 证明如果存在具有性质 (CZ) 的特征函数序列$\{y_{n_{j},r}(x,\lambda_{n_{j},r})\}_{j=1}^{\infty }$, 那么矩阵 $Q$ 是可同时对角化的.

关键词: 向量型 Sturm-Liouville 问题, 重数, 特征值估计, 逆结点问题

Abstract:

The $m$-dimensional vectorial Sturm-Liouville problem with Dirichlet boundary conditions on $(0,1)$ is studied. We firstly discuss the relationship between the matrix-valued potential and the multiplicities of eigenvalues. We prove that if the multiplicities of eigenvalues of $\int_{0}^{1}Q(x){\rm d}x$ are at most $k$ $(1\leq k\leq m-1)$, with finitely many exceptions, the multiplicities of eigenvalues of the vectorial problem are also at most $k$. Then, the inverse nodal problem is investigated with a different method. We show that if there exists an infinite eigenfunctions sequence $\{y_{n_{j},r}(x,\lambda_{n_{j},r})\}_{j=1}^{\infty }$ which are all vectorial functions of type $(CZ)$, then $Q$ is simultaneously diagonalizable.

Key words: Vectorial Sturm-Liouville problems, Multiplicities, Estimation of eigenvalues, Inverse nodal problem

中图分类号:

O175.3

刘肖云,史国良,闫军. 向量型 Sturm-Liouville问题的特征值重数及逆结点问题[J]. 数学物理学报, 2023, 43(3): 669-679.

Liu Xiaoyun,Shi Guoliang,Yan Jun. The Multiplicities of Eigenvalues and Inverse Nodal Problem of a Vectorial Sturm-Liouville Problem[J]. Acta mathematica scientia,Series A, 2023, 43(3): 669-679.

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