Threshold effects of the two-particle Schr\"odinger operators on lattices

We consider a wide class of the two-particle Schrodinger operators $H_{\mu}(k)=H_{0}(k)+\mu V, \,\mu>0,$ with a fixed two-particle quasi-momentum $k$ in the $d$ -dimensional torus $\mathbb{T}^d$, associated to the Bose-Hubbard hamiltonian $H_{\mu}$ of a system of two identical quantum-mechanical particles (bosons) on the $d$- dimensional hypercubic lattice $\mathbb{Z}% ^d$ interacting via short-range pair potentials. We study the existence of eigenvalues of $H_{\mu}(k)$ below the threshold of the essential spectrum depending on the interaction energy $\mu>0$ and the quasi-momentum $k\in \mathbb{T}^d$ of particles. We prove that the threshold (bottom of the essential spectrum), as a singular point (a threshold resonance or a threshold eigenvalue), creates eigenvalues below the essential spectrum under perturbations of both the coupling constant $\mu>0$ and the quasi-momentum $k$ of the particles. Moreover, we show that if the threshold is a regular point, then it does not create any eigenvalues under small perturbations of the coupling constant $\mu>0$ and the quasi-momentum $k$.