We consider the two-particle discrete Schrödinger operator H with a periodic potential perturbed by an exponentially decreasing interaction potential. This operator can be considered as the Hamiltonian of the two-magnon states of ferromagnets with periodically arranged impurities. The operator H can be naturally decomposed in the direct integral of spaces that is related to the analogous direct integral for the periodic operator. We show that the essential spectrum of H in the cell coincides with the band spectrum of the corresponding periodic operator. It is proved that for sufficiently small coupling constants there exists a unique quasi-level (an eigenvalue or a resonance) near the non-degenerate stationary points of eigenvalues of the periodic Schrödinger operator with respect to the chosen component of the quasimomentum. The asymptotic behavior of these quasi-levels for the coupling constant tending to zero is investigated. We obtain the simple sufficient condition when a quasi-level is an eigenvalue.
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