摘要
该文研究了由向量微分表达式:Au(x)=∑_(k=0)^(n)(-1)^(n)(Pk(x)u^((k))(x))^((k)),x∈[x,+∞)产生的自伴向量微分算子.首先,通过引理2.1和引理2.2得到两个向量不等式,利用算子分解定理,分别研究了当系数矩阵Pk(α),k=0,1,…,n为m×m阶实对称正定矩阵和m×m阶实正定对角矩阵时,这类高阶自伴向量微分算子谱的离散性,得到了这类算子谱离散的充分条件,但是必要条件难以给出;其次,作为一个特例,作者研究了只有两项的向量微分算子Au(x)=-(P(a)u^((n))(x))^((n))+Q(x)u(x),u(x)∈C^(0)^(∞)((0,∞),C^(m)),x∈[0,+∞).得到了这类算子的谱是离散的一个充分必要条件,并把这个结论应用到向量值Sturm-Liouville算子和向量值Schrodinger算子,得到了这两类算子的谱离散的充分必要条件;最后,研究了2n阶单项自伴向量微分算子,得到了该类算子谱离散的充分必要条件.
This paper deals with the vector differential operators generated by vectorial dif-ferential expression Au(x)=∑_(k=0)^(n)(-1)^(n)(Pk(x)u^((k))(x))^((k)),x∈[x,+∞).First,we obtain two k=0 vector inequality in Lemma 2.1 and Lemma 2.2,by using operator decomposition theorem,when the coefficient matrix Pk(α),k=0,1,…,n is anm×m order real symmetric positive definite matrix and an order real symmetric positive definite diagonal matrix respectively,the dispersion of the spectrum of the class of higher order self-adjoint vector differential operators is studied,some sufficient conditions for the spectrum of this kind of operators to be discrete are obtained;The second,in the special case,the vector differential operator with only two terms Au(x)=-(P(a)u^((n))(x))^((n))+Q(x)u(x),u(x)∈C^(0)^(∞)((0,∞),C^(m)),x∈[0,+∞).is discussed,the smallest operator generated in its self-adjoint domain is the self-adjoint operator,the sufficient and necessary condition for the spectrum of the kind of operator to be discrete is given;The third,by applying this conclusion to vector-valued Sturm-Liouville operators and vector-valued Schrodinger operators,the necessary and sufficient conditions for spectral dispersion of these two types of operators are obtained.The last,the 2n-th-order mono-term self-adjoint vector differential operator is considered,The necessary and sufficient condition that the spectrum of this kind of operator is discrete is obtained.
作者
钱志祥
Qian Zhixiang(The Department of Basic Education,Guangdong Polytechnic College,Guangdong Zhaoqing 526100,China)
出处
《数学物理学报（A辑）》
CSCD
北大核心
2023年第4期1009-1023,共15页
Acta Mathematica Scientia
基金
广东省教育厅自然基金项目(2019KTSCX248,2021KTSCX157)。
关键词
自伴向量微分算子
自伴扩张
剩余谱
本质谱
离散谱
预紧
Self-adjoint vector differential operator
Self-adjoint extension
Residual spectrum
Essential spectrum
discrete spectrum
Precompact
