Below-Threshold Effects for the Two Particle Discrete Schrödinger Operator on a Lattice

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 $$\partial {{\mathcal{C}}_{j}}$$ , $$j = 0,\,\,2$$ .