The Maslov Index and Spectral Counts for Linear Hamiltonian Systems on $${\mathbb {R}}$$

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.