We consider the family of Schrödinger operators {H } (K), which are associated with the Hamiltonian of a system of two identical bosons on the d -dimensional lattice {Z^d}, where d 3, with interactions on each site and between nearest-neighbor sites with strengths R- and {R--}, respectively. Here, K {T^d} is a fixed quasi-momentum of the particles. We first partition the (, ) - plane into connected components {S₀}, {S₁}, and {C₉}, j = 0, 1, 2. Further, we establish below-threshold effects for {H } (0) on the boundaries of the connected components {S₀} and {C₉}, j = 0, \, \, 2.
Bozorov et al. (Mon,) studied this question.