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

TL;DR

The paper extends Morse index theory to the $n$-dimensional Sturm–Liouville operator on $\mathbb{R}$ by reformulating the problem as a Hamiltonian system and linking the Morse index $m^-(L)$ to a Maslov-type invariant via the CLM index. The authors establish a lower bound for $m^-(L)$ in terms of intersections of unstable subspaces with the Dirichlet Lagrangian, using a careful spectral-flow/homotopy argument and positive crossing forms. This yields an instability criterion for traveling waves in gradient reaction–diffusion systems, whereby the Morse index provides a computable certificate of instability through the zeros of the traveling wave's velocity $\dot u^*$. The work synthesizes CLM Maslov theory, triple/Hörmander indices, and spectral flow to extend classical Morse theory to multi-dimensional SL operators on $\mathbb{R}$ and to practical stability analyses.

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.

Theorems & Definitions (25)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5