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We show that for Sturm-Liouville Systems on the half-line [0, ∞), 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 L2 ((0, ∞), Cn) solutions to the Sturm-Liouville System, and the case in which 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 Hörmander'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 Schrödinger equation on a star graph is linearized about a half-soliton solution. [ABSTRACT FROM AUTHOR]

We describe relations between the Evans function, a modern tool in the study of stability of traveling waves and other patterns for PDEs, and the classical Weyl-Titchmarsh function for singular Sturm-Liouville differential expressions and for matrix Hamiltonian systems. Also, for the scalar Schrödinger equation, we discuss a related issue of approximating eigenvalue problems on the whole line by that on finite segments [ABSTRACT FROM AUTHOR]