单项J-自伴向量微分算子谱的离散性与其系数的关系  

Relationship between discreteness of spectrum and coefficient for mono-term J-self-adjoint vector differential operator

作　　者：钱志祥 林秋红 QIAN Zhi-xiang;LIN Qiu-hong(Department of Basic Courses,Guangdong Polytechnic College,Zhaoqing 526100,Guangdong,China)

机构地区：[1]广东理工学院基础教学部,广东肇庆526100

出　　处：《西北师范大学学报（自然科学版）》2023年第6期13-21,共9页Journal of Northwest Normal University(Natural Science)

基　　金：广东省教育厅自然科学基金资助项目(2019KTSCX248,2021KTSCX157)。

摘　　要：研究二阶单项J-对称向量微分表达式在其J-自伴域内生成的J-自伴向量微分算子L的谱的离散性与其系数之间的关系.当系数矩阵的实部和虚部均是正定对角矩阵时,给出了算子L的所有J-自伴扩张算子的谱离散的充分条件;而当系数矩阵的实部和虚部均是正定对称矩阵时,也得到了这类算子的谱离散的充分条件.利用这两个结论,得到了这类算子谱离散性对其系数依赖的充分条件.最后,对系数矩阵的特征根函数附加更多限制条件,给出了这类算子谱的离散性的一个充要条件.This paper is devoted to studying the relationship between the discreteness of the spectrum of J-self-adjoint vector differential operator L generated by second-order mono-term J-symmetric vector differential expression and its coefficient.When the real part and imaginary part of the coefficient matrix are positive definite diagonal matrix,a sufficient condition that the spectrum of all J-self-adjiont operator of L is discrete is given;when the real part and imaginary part of the coefficient matrix are positive definite symmetric matrix,a similar sufficient condition that the spectrum of this kind of the operator is discrete is obtained.Then,using these two conclusions,some other sufficient conditions for the dependence of the discreteness of the spectrum of this kind of operators on their coefficient are given.Finally,by attaching more restrictions to the eigenroot function of the coefficient matrix,a necessary and sufficient condition that the spectrum of this kind of operator is discrete is also obtained.

关 键 词：J-自伴向量微分算子 单项 离散谱 本质谱 正定矩阵 

分 类 号：O175.3[理学—数学]