Spectral properties of a two-particle hamiltonian on a $d$-dimensional lattice

A system of two arbitrary quantum particles moving on $d$-dimensional lattice interacting via some attractive potential is considered. The number of eigenvalues of the family $h(k)$ is studied depending on the interaction energy of particles and the total quasi-momentum $k\in\mathbb{T}^d$ ($\mathbb{T}^d$ – $d$-dimensional torus). Depending on the interaction energy, the conditions for $h(0)$ that has simple or multifold virtual level at 0 are found.