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

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