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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 $\mu 0$ potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.

Existence of an isolated band in a system of three particles in an optical lattice