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一类不连续Sturm−Liouville问题的特征值与特征函数零点估计

Esimation of Eigenvalues and Zeros of Eigenfunction for a Class of Discontinuous Sturm−Liouville Problems

  • 摘要: 针对带有周期边界条件的不连续Sturm−Liouville问题(SLPs),利用常微分方程初值理论计算出不连续SLPs的2个线性无关解的渐进估计;利用Gronwall不等式、微分方程特征值性质及解的渐进估计式推导出不连续SLPs问题特征值的渐进估计形式;利用Prufer变换得到不连续SLPs问题第 n 个特征值对应的特征函数在 \left(0,c\right)\cup \left(c,\textπ\right) 内有 n 个零点。本文研究对于计算不连续SLPs特征值下标及研究解的振荡性有重要意义。

     

    Abstract: For discontinuous Sturm-Liouville problems (SLPs) with periodic boundary conditions, utilizing the theory of initial value problems for ordinary differential equations, asymptotic estimates for two linearly independent solutions of the aforementioned discontinuous SLPs were calculated. By leveraging Gronwall’s inequality, properties of eigenvalues in differential equations, and the asymptotic estimation formula of the solutions, an asymptotic estimation form for the eigenvalues of the discontinuous SLPs was derived.Using the Prufer transformation, it was shown that the nth eigenfunction corresponding to the nth eigenvalue of the discontinuous SLPs has n zeros within the interval (0,c)∪(c,π).These findings are significant for calculating the indices of eigenvalues in discontinuous SLPs and for studying the oscillatory behavior of their solutions.

     

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