一类具有Neumann边界条件的向量型Sturm-Liouville问题的谱性质

Spectral Properties of a Class of Vector Sturm-Liouville Problems with Neumann Boundary Conditions

摘要: 本文在势函数$Q\left(x\right)$ 为一般的实对称矩阵的前提下，研究了Neumann边界条件下的向量型Sturm-Liouville问题的特征值的重数问题。讨论了特征值及特征函数判别式的渐近式，并给出了关于特征值重数的重要结论：如果矩阵${\displaystyle {\int }_0^{1}Q}\left(\xi \right)d\xi$ 的特征值重数至多为$k\left(1\le k\le m-1\right)$ ，那么，除了有限个特征值，向量型Sturm-Liouville问题的特征值重数也至多为$k$ 。

Abstract: This paper investigates the multiplicity of eigenvalues for vector-valued Sturm-Liouville problems under Neumann boundary conditions, with the potential function $Q\left(x\right)$ being a general real symmetric matrix. The asymptotic expressions of eigenvalues and characteristic functions are discussed, and the significant conclusion regarding the multiplicity of eigenvalues are presented: If the multiplicity of eigenvalues of the matrix ${\displaystyle {\int }_0^{1}Q}\left(\xi \right)d\xi$ is at most $k\left(1\le k\le m-1\right)$ , then, except for finitely many eigenvalues, the multiplicity of eigenvalues for the vector-valued Sturm-Liouville problem is also at most $k$ .