  We show that for Sturm-Liouville Systems on the half-line $[0,\infty)$, the
Morse index can be expressed in terms of the Maslov index and an additional
term associated with the boundary conditions at $x = 0$. Relations are given
both for the case in which the target Lagrangian subspace is associated with
the space of $L^2 ((0,\infty), \mathbb{C}^{n})$ solutions to the
Sturm-Liouville System, and the case when the target Lagrangian subspace is
associated with the space of solutions satisfying the boundary conditions at $x
= 0$. In the former case, a formula of H\"ormander's is used to show that the
target space can be replaced with the Dirichlet space, along with additional
explicit terms. We illustrate our theory by applying it to an eigenvalue
problem that arises when the nonlinear Schr\"odinger equation on a star graph
is linearized about a half-soliton solution.
