Existence of bound states of $N$-body problem in an optical lattice

We provide sufficient conditions to have at least one $N$-particle bound state below the essential spectrum of a large class of $N$-particle discrete Schr"odinger operators $H(K),$ $K\in {\mathbb{T}}^d,$ $d\ge 1,$ associated to the Hamiltonian of (not necessarily identical) $N$ particles, moving on a lattice ${\mathbb{Z}}^d$ and interacting via short-range pair potentials. We also describe the essential spectrum of $H(K)\mathrm{.}$