Number of bound states of the Hamiltonian of a lattice two-boson system with interactions up to the next neighbouring sites

We study the family Hγ λ μ(K), K∈ T², of discrete Schr\"odinger operators, associated to the Hamiltonian of a system of two identical bosons on the two-dimen\-sional lattice Z², interacting through on one site, nearest-neighbour sites and next-nearest-neighbour sites with interaction magnitudes γ,λ and μ, respectively. We prove there existence an important invariant subspace of operator Hγ λ μ(0) such that the restriction of the operator Hγ λ μ(0) on this subspace has at most two eigenvalues lying both as below the essential spectrum as well as above it, depending on the interaction magnitude λ,μ∈ R (only). We also give a sharp lower bound for the number of eigenvalues of Hγλμ(K).