The Threshold effects for the two-particle Hamiltonians on lattices
  For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice $\bbZ^3$, $k$ being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its threshold is non-empty. The assumptions are:   (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free   Hamiltonians in the coordinate representation generate positivity preserving semi-groups. 
Sergio Albeverio, Saidakhmat N. Lakaev, Konstantin A. Makarov and   Zahriddin I. Muminov