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

Abstract

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 $H(K),$ $K\in \mathbb{T}^d,$ $d\ge1,$ 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 $H(K).$