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

For a wide class of two-particle Schrödinger operators $H(k)=H_0(k)+V$, $k\in\mathbb T^d$, corresponding to a two-fermion system on a $d$-dimensional cubic integer lattice $(d\ge1)$, we prove that for any value $k\in\mathbb T^d$ 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) Schrödinger operator in the coordinate representation generates a semigroup that preserves positivity.