Scattering by a periodic tube in R3: part i. The limiting absorption principle*Dedicated to the memory of my friend and colleague Armin Lechleiter. (9th September 2019) - Kirsch, Andreas
Abstract: Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium, described by a refractive index which is periodic along the axis of an infinite cylinder in and constant outside of the cylinder. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. By the standard one-dimensional Floquet–Bloch transform and the introduction of the exterior Dirichlet–Neumann map we first reduce the scattering problem to a class of periodic problems in a bounded cell, depending on the wave number k and the Bloch parameter . We use a functional analytic singular perturbation result to study this problem in a neighborhood of a singular pair . This abstract result allows us to derive explicitly a representation for the limiting absorption solution as a sum of a decaying part (along the axis of the cylinder) and a finite sum of propagating modes.