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We present a detailed study of the scattering system given by the NeumannLaplacian on the discrete half-space perturbed by a periodic potential at theboundary. We derive asymptotic resolvent expansions at thresholds andeigenvalues, we prove the continuity of the scattering matrix, and we establishnew formulas for the wave operators. Along the way, our analysis puts intoevidence a surprising relation between some properties of the potential, likethe parity of its period, and the behaviour of the integral kernel of the waveoperators.

Discrete Laplacian in a half-space with a periodic surface potential I: Resolvent expansions, scattering matrix, and wave operators