Abstract
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 \(H_{\gamma\lambda\mu}(K)\) .
