A two-boson lattice Hamiltonian with interactions up to next-neighboring sites

A system of two identical spinless bosons on the two-dimensional lattice is considered under the assumption that on-site and first and second nearest-neighboring site interactions between the bosons are only nontrivial and that these interactions are of magnitudes $\gamma$, $\lambda$, and $\mu$, respectively. A partition of the $(\gamma,\lambda,\mu)$-space into connected components is established such that, in each connected component, the two-boson Schroedinger operator corresponding to the zero quasi-momentum of the center of mass has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential (continuous) spectrum and above its top. Moreover, for each connected component, a sharp lower bound is established on the number of isolated eigenvalues for the two-boson Schr\"odinger operator corresponding to any admissible nonzero value of the center-of-mass quasimomentum.