向量型Sturm-Liouville问题的特征值重数及逆结点问题  

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

作　　者：刘肖云[1] 史国良 闫军 Liu Xiaoyun;Shi Guoliang;Yan Jun(School of Mathematics and Information Science,Anyang Institute of Techology,Henan Anyang 455000;School of Mathematical Sciences,Tianjin University,Tianjin 300354)

机构地区：[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

出　　处：《数学物理学报（A辑）》2023年第3期669-679,共11页Acta Mathematica Scientia

基　　金：国家自然科学基金(12001153,62065015);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)资助~~。

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∫_(0)1Q(x)dx的特征值重数至多为k(1≤k≤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_(nj,r)(x,λ_(nj,r))}_(j)^(∞)=1,那么矩阵Q是可同时对角化的.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∫_(0)1Q(x)dx are at most k(1≤k≤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_(nj,r)(x,λ_(nj,r))}_(j)^(∞)=1 which are all vectorial functions of type(CZ),then Q is simultaneously diagonalizable.

关 键 词：向量型 STURM-LIOUVILLE 问题 重数 特征值估计 逆结点问题 

分 类 号：O175.3[理学—数学]