Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of the family of the 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). Depending on the particle interaction energy, we obtain conditions under which the left edge of the continuous spectrum is simultaneously a multiple virtual level and an eigenvalue of the operator $h(\mathbf 0)$.