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-neighbour sites and next-nearest-neighbour sites with interaction magnitudes \(\gamma,\lambda\) and \(\mu,\) respectively. We prove there existence an invariant subspace of the operator \(H_{\gamma\lambda\mu}(0)\) that its restriction on this subspace has only one simple eigenvalue, which lay below or above of its essential spectrum depending on the interaction magnitude \(\mu\in\mathbb{R}\) . Applying this result we give a lower bound for the number of the discrete eigenvalues of the operator \(H_{\gamma\lambda\mu}(K)\) for all \(K\in\mathbb{T}^{2}\) .
