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

TL;DR

This paper establishes conditions under which N-particle bound states exist in an optical lattice, analyzing the spectrum of discrete Schrödinger operators for particles with short-range interactions.

Contribution

It provides new sufficient conditions for the existence of bound states and characterizes the essential spectrum of N-particle lattice Hamiltonians.

Findings

Bound states exist below the essential spectrum under certain conditions.

The essential spectrum of the N-particle Hamiltonian is explicitly described.

Results apply to particles with non-identical interactions on a 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 $H(K).$