Threshold effects in a two-fermion system on an optical lattice

For a wide class of two-particle Schrodinger operators H(k) = H-0(k) + V, k is an element of Td, corresponding to a two-fermion system on a d-dimensional cubic integer lattice (d >= 1), we prove that for any value k is an element of Td of the quasimomentum, the discrete spectrum of H(k) below the lower threshold of the essential spectrum is a nonempty set if the following two conditions are satisfied. First, the two-particle operator H(0) corresponding to a zero quasimomentum has either an eigenvalue or a virtual level on the lower threshold of the essential spectrum. Second, the one-particle free (nonperturbed) Schrodinger operator in the coordinate representation generates a semigroup that preserves positivity.