| 摘要: |
| 本文在势函数~$Q(x)$~为一般实对称矩阵的前提下, 研究了一般分离边界条件下的~$m$~阶向量型~Sturm-Liouville~问题的特征值重数问题. 利用特征值及特征函数判别式的渐近式, 针对特征值重数问题, 获得了重要结论: 若矩阵~$\int^1_0Q(\xi)d\xi$~的特征值的重数不超过~$k(1\leq k\leq m-1)$, 那么, 除了有限个特征值, 向量型~Sturm-Liouville~问题的特征值重数也不超过~$k$. |
| 关键词: 向量型~Sturm-Liouville~问题 重数 特征值估计 |
| DOI： |
| 分类号:O175 |
| 基金项目: |
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| THE SPECTRAL PROPERTIES OF A CLASS OF VECTORIAL STURM-LIOUVILLE PROBLEMS WITH GENERAL SEPARATED BOUNDARY CONDITIONS |
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wangyijing, GaoYunlan
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| Abstract: |
| This paper studies the eigenvalue multiplicity problem for vector-valued Sturm-Liouville problems of order $m$ with general separated boundary conditions, under the assumption that the potential function $Q(x)$ is a general real symmetric matrix. Using asymptotic expressions for the eigenvalue and eigenfunction discriminants, an important conclusion regarding eigenvalue multiplicity is obtained: if the multiplicity of the eigenvalues of the matrix $\int^1_0Q(\xi)d\xi$ does not exceed $k$ $(1 \leq k \leq m-1)$, then, except for finitely many eigenvalues, the multiplicity of the eigenvalues of the vector-valued Sturm-Liouville problem also does not exceed $k$. |
| Key words: Vectorial Sturm-Liouville problems Multiplicity Eigenvalue estimation. |
