Abstract
We consider the family of Schrödinger operators \({{H}_{{\gamma \lambda }}}(K)\) , which are associated with the Hamiltonian of a system of two identical bosons on the \(d\) -dimensional lattice \({{\mathbb{Z}}^{d}}\) , where \(d \geqslant 3\) , with interactions on each site and between nearest-neighbor sites with strengths \(\gamma \in \mathbb{R}_{{\text{-}}}\) and \(\lambda \in {{\mathbb{R}}_{{\text{--}}}}\) , respectively. Here, \(K \in {{\mathbb{T}}^{d}}\) is a fixed quasi-momentum of the particles. We first partition the \((\gamma ,\lambda ) - \) plane into connected components \({{\mathcal{S}}_{0}},\) \({{\mathcal{S}}_{1}}\) , and \({{\mathcal{C}}_{j}}\) , \(j = 0,1,2\) . Further, we establish below-threshold effects for \({{H}_{{\gamma \lambda }}}(0)\) on the boundaries of the connected components \(\partial {{\mathcal{S}}_{0}}\) and \(\partial {{\mathcal{C}}_{j}}\) , \(j = 0,\,\,2\) .
