  We present a detailed study of the scattering system given by the Neumann
Laplacian on the discrete half-space perturbed by a periodic potential at the
boundary. We derive asymptotic resolvent expansions at thresholds and
eigenvalues, we prove the continuity of the scattering matrix, and we establish
new formulas for the wave operators. Along the way, our analysis puts into
evidence a surprising relation between some properties of the potential, like
the parity of its period, and the behaviour of the integral kernel of the wave
operators.
