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We consider the spectrum of the discrete Schr\"odinger equation withone-dimensional perturbation. We obtain the explicit form of scattering matrixand find the exact condition of absence of singular part of the spectrum. Wecalculated also the eigenvalue that appears if this condition is not true. Inthe last part of our paper we give few remarks on the case of two-dimensionalperturbations.

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