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

2025/04/07 by Ran Yang, Yang, Ran, Qin Xing +1
Computer Science · Engineering · Mathematics · #Control and Stability of Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2504.05091

openalex publication_date 2025/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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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