一类高阶自伴向量微分算子谱的离散性及其应用  

Discreteness of the Spectrum of a Class of Higher Order Self-adjoint Vector Differential Operators and its Application

作　　者：钱志祥 Qian Zhixiang(The Department of Basic Education,Guangdong Polytechnic College,Guangdong Zhaoqing 526100,China)

机构地区：[1]广东理工学院基础教学部,广东肇庆526100

出　　处：《数学物理学报（A辑）》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)。

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.

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 

分 类 号：O175.3[理学—数学]