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数学进展 - 2020, Vol. 49(4): 443-452
研究论文
正则型Sturm-Liouville微分算子特征值关于边界条件的连续依赖性
Continuous Dependence of Eigenvalues of Regular Sturm-Liouville Differential Operators on the Boundary Condition

杨昕雅
YANG Xinya

陕西师范大学数学与信息科学学院, 西安, 陕西, 710119
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi, 710119, P. R. China

收稿日期: 2019-06-10
出版日期: 2020-08-11
2020, Vol. 49(4): 443-452
DOI: 10.11845/sxjz.2019065b


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摘要 本文以隐函数存在定理为主要工具, 重新研究Sturm-Liouville微分算子特征值关于边界条件参数的连续依赖性问题. 我们不仅给出了该结果一个简单的新证明, 而且明确地呈现了第$n$个特征值关于边界条件参数的导数,进而得到了在实耦合型边界条件下二重特征值产生的位置及个数的新结果.
关键词 正则Sturm-Liouville算子特征值隐函数存在定理    
Abstract:In this paper, we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem. The work not only provides a new and elementary proof of the above results, but also explicitly presents the expressions for derivatives of the $n$-th eigenvalue with respect to given parameters. Furthermore, we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.
Key wordsregular Sturm-Liouville operator    eigenvalue    implicit function theorem
PACS:  O175.3  
基金资助:国家自然科学基金(No. 11571212).
通讯作者: E-mail: yangxinya@snnu.edu.cn   
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