The existence of an isolated band in a system of three particles in an optical lattice

We prove the existence of two-and three-particle bound states of the Schr\"odinger operators $h_\mu(k),k\in \T^d$ and $H_\mu(K),K\in \T^d$ associated to Hamiltonians $\mathrm{h}_{\mu}$ and $\mathrm{H}_{\mu}$ of a system of two and three identical bosons on the lattice $\Z^d, d=1,2$ interacting via pairwise zero-range attractive $\mu<0$ or repulsive $\mu>0$ potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.