| 刘肖云,史国良,闫军.自伴向量型Sturm-Liouville问题特征值λn,r的依赖性*[J].数学年刊A辑,2021,42(3):289~304 |
| 自伴向量型Sturm-Liouville问题特征值λn,r的依赖性* |
| Dependence of the Eigenvalue λn,r of Self-Adjoint Vectorial Sturm-Liouville Problems |
| Received:May 27, 2019 Revised:March 05, 2021 |
| DOI：10.16205/j.cnki.cama.2021.0023 |
| 中文关键词: 向量型Sturm-Liouville问题, 矩阵Pr¨ufer变换, 特征值, 特征函数零点, 连续依赖性 |
| 英文关键词:Vectorial Sturm-Liouville problem, Matrix Pr¨ufer transformation,Eigenvalues, Zeros of eigenfunctions, Continuous dependence |
| 基金项目:国家自然科学基金(No.11801012), 天津市青年基金(No.20JCQNJC01440), 河南省科技攻关项目(No.212102310383)和河北省自然科学基金(No.A2019202205) |
| Author Name | Affiliation | | LIU Xiaoyun | School of Mathematics and Information Science, Anyang Institute of Technology, Anyang 300072, Henan, China. | | SHI Guoliang | School of Mathematics, Tianjin University, Tianjin 210000, China. | | YAN Jun | School of Mathematics, Tianjin University, Tianjin 210000, China. |
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| 中文摘要: |
| 研究定义在区间[a,b]上的m维自伴向量型Sturm-Liouville问题.首先, 利用矩阵Pr¨ufer变换讨论该问题特征值的分布, 同时得到第n组特征值λn,r(n ∈ N0, r = 1, 2, · · · , m)所对应的特征函数un,r(x)在区间(a,b)内恰有n个零点.然后, 研究了特征值λn,r分别关于算子系数和边界条件的连续依赖性. 在此基础上, 假设所有特征值都是单重的,建立了第n组特征值λn,r (r = 1, 2, · · · , m)关于首项系数P-1, 势矩阵Q, 权矩阵W的微分表达式,进而讨论特征值关于P-1, Q, W的单调性. 最后, 如果允许特征值的指标可以跳跃,则任一特征值都可以嵌入到一个连续的特征值分支中,从而证明λn,r关于边界条件中的参数α和β的连续可微性. |
| 英文摘要: |
| Regular m-dimensional vectorial Sturm-Liouville problem on the interval [a, b]is considered. Firstly, the authors study the eigenvalues of the problem using the tool of generalized matrix Pr¨ufer transformation. They derive that the number of zeros of eigenfunction un,r(x) corresponding to λn,r (n = 0, 1, · · · ; r = 1, 2, · · · , m) is exactly n. Secondly,continuous dependence of the eigenvalue λn,r on the equation and on the boundary conditions is studied separately. Then an expression for the derivative of eigenvalue λn,r with respect to a given parameter: a leading coefficient matrix, a potential matrix and a weight matrix, is found under the assumption that λn,r are all simple. Meanwhile, the authors show the monotonicity of eigenvalues in these parameters. Lastly, they show the fact that if the indices of λn,r are allowed to jump, then each eigenvalue can be embedded in an eigenvalue branch which is not only continuous but differentiable on the boundary conditions. |
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