  This paper continues the analysis of the scattering problem for a network of
open wave-guides started in [arXiv:2302.04353, arXiv:2310.05816]. In this part
we present explicit, physically motivated radiation conditions that ensure
uniqueness of the solution to the scattering problem. These conditions stem
from a 2000 paper of A. Vasy on 3-body Schrodinger operators; we discuss
closely related conditions from a 1994 paper of H. Isozaki. Vasy's paper also
proves the existence of the limiting absorption resolvents, and that the
limiting solutions satisfy the radiation conditions. The statements of these
results require a calculus of pseudodifferential operators, called the 3-body
scattering calculus, which is briefly introduced here. We show that the
solutions to the model problem obtained in arXiv:2302.04353 satisfy these
radiation conditions, which makes it possible to prove uniqueness, and
therefore existence, for the system of Fredholm integral equations introduced
in that paper.
