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

Ran Yang; Qin Xing

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arXiv · 2504.05091

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

Ran Yang·Qin Xing

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Abstract

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.

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Ran Yang, Qin Xing. 2025-04-07. Morse Index Theorem for Sturm-Liouville Operators on the Real Line. https://arxiv.org/abs/2504.05091

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