The Hamiltonian of a system of four arbitrary quantum particles with three-particle short-range interaction potentials on a three-dimensional lattice is examined. The location of the essential spectrum of this Hamiltonian is described by Faddeev’s equations.
We prove the existence of two-and three-particle bound states of the Schrödinger operators $h_μ(k),k\in \T^d$ and $H_μ(K),K\in \T^d$ associated to Hamiltonians hμ and Hμ of a system of two and three identical bosons on the lattice Zd,d=1,2 interacting via pairwise zero-range attractive μ<0 or repulsive μ>0 potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.