On the spectrum of discrete Schr\"odinger equation with one-dimensional perturbation

We consider the spectrum of the discrete Schr"odinger equation with one-dimensional perturbation. We obtain the explicit form of scattering matrix and find the exact condition of absence of singular part of the spectrum. We calculated also the eigenvalue that appears if this condition is not true. In the last part of our paper we give few remarks on the case of two-dimensional perturbations.