Resonance multiplicity of a perturbed periodic Schrödinger operator

We consider the perturbation of a periodic Schrödinger operator by a potential that is periodic in the variables $x_1$ and $x_2$ and exponentially decreases as $|x_3| \to \infty$. Near the zero surface of the derivative of the eigenvalue of the periodic operator in a cell with respect to the third quasi-momentum component, we obtain relations between the resonance multiplicity and the order of the pole of the quantities characterizing the scattering. As a rule, the forward scattering amplitude vanishes on this surface.