The Limiting Absorption Principle for a Periodic Semi-Infinite Waveguide

We study an elliptic boundary value problem in a semi-infinite, straight waveguide containing a periodic material distribution. The spectral parameter is taken to lie in a spectral band of the corresponding full-space periodic operator. In contrast to the situation with constant coefficients, it is not obvious how to define the “outgoing” solution of the problem. In order to define the outgoing solution, we introduce a small absorption to regularize the problem and then prove that the solutions converge locally in H1.