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

TL;DR

The paper develops physically motivated radiation conditions for open wave-guide networks to guarantee uniqueness of scattering solutions in a high-dimensional, non-decaying-channel setting. It unifies Isozaki’s conic-radiation framework with Vasy’s 3-body scattering calculus to formulate and justify radiation conditions via the scattering wave-front set and to prove the limiting absorption principle in this context. It then applies these tools to a two-dimensional model problem, recasting scattering as a transmission problem and proving uniqueness for the associated integral equations, while also connecting the integral-equation approach with LAP solutions. The results provide a rigorous foundation for existence, uniqueness, and numerics of scattering off complex wave-guide networks and pave the way for extensions to Maxwell systems and more general geometries.

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.

Figures (5)

• Figure 1: Two dielectric channels in $\mathbb R^2$ meeting along a straight interface.
• Figure 2: The upper left panel shows the incoming waveguide mode, $u^{\operatorname{in}},$ the upper right panel shows the total field, $u^{\operatorname{tot}}.$ The lower left panel shows the wave-guide mode part of $u^{\operatorname{out}},$ and the lower right panel the radiation part of $u^{\operatorname{out}}.$ These numerical solutions and plots are obtained using MATLAB software written by Tristan Goodwill and described in GE_2024.
• Figure 3: Three dielectric channels, indicated as tubes around rays extending to infinite, meeting in a compact interaction zone, $\Omega_0.$
• Figure 4: Schematic illustration of the wave-guide compactification of $\mathbb R^2,$ where $\mathcal{C}=\{(\pm 1,0)\}.$ Coordinates are shown near the intersection $\operatorname{mf}\cap \operatorname{ff}_{(1,0)}.$ The red curves are level sets of $s$ and the blue curves are level sets of $\zeta_1.$ As indicated, in this coordinate neighborhood, $\operatorname{ff}_{(1,0)}=\{s=0\},$ and $\operatorname{mf}=\{\zeta_1=0\}.$
• Figure 5: Polar coordinates, $r_{\pm}\eta^{\pm},$ adapted to the locations of the channels.

Theorems & Definitions (29)

• Definition 1
• Definition 2
• Definition 3
• Definition 4: Isozaki's radiation condition
• Remark 1
• Remark 2
• Theorem 1: VasyAsterisque, Proposition 17.8
• Theorem 2: VasyAsterisque, Theorem 18.3
• Definition 5
• Remark 3