The Maslov index and spectral counts for linear Hamiltonian systems on $\mathbb{R}$

TL;DR

This paper establishes a framework linking the Maslov index to spectral counts for linear Hamiltonian systems on the real line, with applications to stability analysis of nonlinear waves.

Contribution

It introduces a general method to relate the Maslov index to eigenvalue counts for various Hamiltonian systems, including Sturm-Liouville and nonlinear spectral parameter systems.

Findings

Framework applicable to Sturm-Liouville systems

Framework applicable to fourth-order potential systems

Framework applicable to nonlinear spectral parameter systems

Abstract

Working with a general class of linear Hamiltonian systems specified on $\mathbb{R}$, we develop a framework for relating the Maslov index to the number of eigenvalues the systems have on intervals of the form $[\lambda_1, \lambda_2)$ and $(-\infty, \lambda_2)$. We verify that our framework can be implemented for Sturm-Liouville systems, fourth-order potential systems, and a family of systems nonlinear in the spectral parameter. The analysis is primarily motivated by applications to the analysis of spectral stability for nonlinear waves, and aspects of such analyses are emphasized.