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 Schrödinger operators $h_{\mu }\left(k\right),k\in T^d$ and $H_{\mu }\left(K\right),K\in T^d$ associated to Hamiltonians $h_{\mu }$ and $H_{\mu }$ 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. (paper)