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 $H_{\gamma \lambda \mu }(K)$, $K\in {\mathbb{T}}^2,$ of discrete Schrödinger operators, associated to the Hamiltonian of a system of two identical bosons on the two-dimen-sional lattice ${\mathbb{Z}}^2,$ interacting through on one site, nearest-neighbour sites and next-nearest-neighbour sites with interaction magnitudes $\gamma ,\lambda$ and $\mu ,$ respectively. We prove there existence an important invariant subspace of operator $H_{\gamma \lambda \mu }(0)$ such that the restriction of the operator $H_{\gamma \lambda \mu }(0)$ 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 $\lambda ,\mu \in \mathbb{R}$ (only). We also give a sharp lower bound for the number of eigenvalues of $H_{\gamma \lambda \mu }(K)$.