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 $d\geq 3$, with interactions on each site and between nearest-neighbor sites with strengths $\gamma \in \mathbb{R}^-$ and $\lambda \in \mathbb{R^-}$, 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 $\partial\mathcal{C}_j, j=0,2$.