Solving the Scattering Problem for Open Wave-Guide Networks, III: Radiation Conditions and Uniqueness

Abstract

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.