标题： 实数线上关于Sturm-Liouville算子的Morse指数定理 显示英文标题

标题： Morse Index Theorem for Sturm-Liouville Operators on the Real Line

摘要： 经典Morse指标定理建立了Morse指标（用于刻画线性自伴微分算子关键谱性质的负特征值个数）与相应共轭点计数之间的基本联系。本文中，我们将这些基础结果推广到$\mathbb{R}$上的Sturm-Liouville算子。特别是，对于自治Lagrangian系统，我们采用几何方法推导出Morse指标的一个下界。作为具体应用，我们建立了一个判据，用于检测梯度反应扩散系统中的行波不稳定性。 显示英文摘要

摘要： The classical Morse index theorem establishes a fundamental connection between the Morse index-the number of negative eigenvalues that characterize key spectral properties of linear self-adjoint differential operators-and the count of corresponding conjugate points. In this paper, we extend these foundational results to the Sturm-Liouville operator on $\mathbb{R}$. In particular, for autonomous Lagrangian systems, we employ a geometric argument to derive a lower bound for the Morse index. As concrete applications, we establish a criterion for detecting instability in traveling waves within gradient reaction-diffusion systems.