The existence of bound states in a system of three particles in an optical lattice

TL;DR

This paper proves the existence of bound states in a three-particle lattice system with two identical fermions and one different particle, for both attractive and repulsive interactions, across different quasi-momenta.

Contribution

It demonstrates the existence of bound states in a three-particle lattice system with zero-range interactions for all nonzero interaction strengths.

Findings

Bound states exist for all nonzero interaction strengths.

Bound states are associated with the three-particle quasi-momentum.

Results hold for both attractive and repulsive potentials.

Abstract

We consider the hamiltonian $\mathrm{H}_{\mu},\mu\in \R$ of a system of three-particles (two identical fermions and one different particle) moving on the lattice ${\Z}^d ,\, d=1,2 $ interacting through repulsive $(\mu>0)$ or attractive $(\mu<0)$ zero-range pairwise potential $\mu V$. We prove for any $\mu\ne0$ the existence of bound state of the discrete three-particle Schr\"odinger operator $H_{\mu}(K),\,K\in \T^d$ being the three-particle quasi-momentum, associated to the hamiltonian $\mathrm{H}_{\mu}$.