Resonance multiplicity of a perturbed periodic Schrodinger operator

We consider the perturbation of a periodic Schrodinger operator by a potential that is periodic in the variables x(1) and x(2) and exponentially decreases as \x3\ --> infinity. 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.