RESONANCE MULTIPLICITY OF A PERTURBED PERIODIC SCHRODINGER OPERATOR

We consider the perturbation of a periodic Schrodinger operator by a p otential that is periodic in the variables x(1) and x(2) and exponenti ally decreases as \x3\ --> infinity. Near the zero surface of the deri vative of the eigenvalue of the periodic operator in a cell with respe ct to the third quasi-momentum component we obtain relations between t he resonance multiplicity and the order of the pole of the quantities characterizing the scattering. As a rule, the forward scattering ampli tude vanishes on this surface.