Number of Bound States of the Hamiltonian of a Lattice Two-boson System with Interactions up to the Next Neighboring 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-dimensional lattice $$\mathbb{Z}^{2},$$ interacting through on one site, nearest-neighbor sites and next-nearest-neighbor 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 $$H_{\gamma\lambda\mu}(K)$$ .