Electron scattering by a crystal layer

We consider the one-particle discrete Schrödinger operator $H$ with a periodic potential perturbed by a function $\varepsilon W$ that is periodic in two variables and exponentially decreasing in the third variable. Here, $\varepsilon$ 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 $\varepsilon W$. We obtain simple formulas for the transmission and reflection probabilities.