Spectral properties of a two-particle Hamiltonian on a lattice

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with some dispersion functions (describing particle transport from a site to a neighboring site). The particles interact via an attractive potential at only the nearest-neighbor sites. We study how the number of eigenvalues of a family of operators $h(k)$ depends on the particle interaction energy and the total quasimomentum $k\in\mathbb T^3$, where $\mathbb T^3$ is a three-dimensional torus. We find the conditions under which the operator $h(\mathbf 0)$ has a double or triple virtual level at zero depending on the particle interaction energy.