On small perturbations of the Schrödinger equation with periodic potential

We consider small perturbations of the potential periodic in variables $x_j$, $j=1,2,3$, by a function wich is periodic in $x_1$, $x_2$ and exponentially decreases as $|x_3|\to\infty$. We prove that close to energies corresponding to the extrema in the third component of the quasy-momentum of nondegenerate eigenvalues of the Schrödinger operator with periodic potential considered in the cell there exists a unique (up to multiplicative factor) solution of the integral equation describing both eigenvalues and resonance levels. The asymptotic behaviour of the latter quantities is described.