Spectral properties of two particle Hamiltonian on one-dimensional lattice

We consider a system of two arbitrary quantum particles on a one-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 $k\in\mathbb T$ (where $\mathbb T$ is a one-dimensional torus). Depending on the particle interaction energy, we obtain conditions for existence of multiple eigenvalues below the essential spectrum of operator $h(k)$.