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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 Schrödinger operators $h_μ(k),k\in \T^d$ and $H_μ(K),K\in \T^d$ associated to Hamiltonians $\mathrm{h}_μ$ and $\mathrm{H}_μ$ of a system of two and three identical bosons on the lattice $\Z^d, d=1,2$ 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.