# The Maslov and Morse Indices for Sturm-Liouville Systems on the Half-Line

## TL;DR

This paper establishes a relationship between the Morse index and the Maslov index for Sturm-Liouville systems on the half-line, incorporating boundary conditions, and applies it to a nonlinear Schrödinger eigenvalue problem.

## Contribution

It provides a new formula linking Morse and Maslov indices for Sturm-Liouville systems with boundary conditions, extending previous results and illustrating with a Schrödinger eigenvalue application.

## Key findings

- Morse index expressed via Maslov index and boundary terms
- Formulas valid for solutions in L^2 and boundary condition spaces
- Application to eigenvalue problem in nonlinear Schrödinger equation

## Abstract

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.