In this paper, we discuss the continuous dependence of eigenvalue of Sturm-Liouville operators on the real coupled boundary condition by using of implicit function theorem. A geometric structure on containing real coupled boundary conditions is firstly clarified, that is, the smooth embedding submanifold. Under this structure, we verify the continuous differentiability of the -th eigenvalue with regard to the boundary condition and explicitly present the expression for its differential. Moreover, a sufficient condition for recognizing double eigenvalues is given.

中文翻译：

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Sturm-Liouville 算子特征值对实耦合边界条件的依赖性

本文利用隐函数定理讨论了Sturm-Liouville算子特征值对实耦合边界条件的连续依赖性。首先明确了包含实耦合边界条件的几何结构，即光滑嵌入子流形。在这种结构下，我们验证了第-个特征值对于边界条件的连续可微性，并明确地给出了其微分的表达式。此外，还给出了识别双特征值的充分条件。