On eigenvalues and virtual levels of a two-particle Hamiltonian on a $d$-dimensional lattice

The two-particle Schrödinger operator $h_\mu(k)$, $k\in\mathbb{T}^d$ (where $\mu>0$, $\mathbb{T}^d$ is a $d$-dimensional torus), associated to the Hamiltonian h of the system of two quantum particles moving on a $d$-dimensional lattice, is considered as a perturbation of free Hamiltonian $h_0(k)$ by the certain $3^d$ rank potential operator $\mu\mathbf{v}$. The existence conditions of eigenvalues and virtual levels of $h_\mu(k)$, are investigated in detail with respect to the particle interaction $\mu$ and total quasi-momentum $k\in\mathbb{T}^d$.