Electron scattering by a crystal layer

We consider the one-particle discrete Schrödinger operator H with a periodic potential perturbed by a function ?W that is periodic in two variables and exponentially decreasing in the third variable. Here, ? is a small parameter. We study the scattering problem for H near the point of extremum with respect to the third quasimomentum coordinate for a certain eigenvalue of the Schrödinger operator with a periodic potential in the cell, in other words, for the small perpendicular component of the angle of particle incidence on the potential barrier ?W. We obtain simple formulas for the transmission and reflection probabilities.