The Threshold Effects for the Two Particle Discrete Schrödinger Operators on Lattices

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.