Abstract
We study the Schrödinger operators \({H}_{\gamma\lambda}(K)\) , with \(K\in\mathbb{T}^{d},\) the fixed quasimomentum of the particles pair, associated with a system of two identical bosons on the \(d\) -dimensional lattice \(\mathbb{Z}^{d},\,d\geqslant 3\) with on one site and on nearest-neighboring-site interactions of magnitudes \(\gamma\in\mathbb{R}\) and \(\lambda\in\mathbb{R}\) , respectively. We partition the \((\gamma,\lambda)-\) plane into connected components such that, in each connected components the number of eigenvalues of the Schrödinger operator \({H}_{\gamma\lambda}(0)\) remains constant. Moreover, we establish that the operator \({H}_{\gamma\lambda}(0)\) has in each boundary of the connected components either a threshold eigenvalue or a threshold resonance. We also find a sharp lower bound for the number of isolated eigenvalues of \({H}_{\gamma\lambda}(K)\) overall \(K\in\mathbb{T}^{d}\) , on each boundary of the connected components.
