  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.
