The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice

We consider the Hamiltonian $\hat h_{\mu\lambda}$, $\mu,\lambda\ge0$, describing the motion of one quantum particle on a three-dimensional lattice in an external field. We investigate the number of eigenvalues and their arrangement depending on the value of the interaction energy for $\mu\ge0$ and $\lambda\ge0$.