Integral equations for the four-body problem

TL;DR

This paper develops an integral equation approach for analyzing four-body quantum problems involving bosons and fermions with Feshbach resonances, simplifying the complex interactions into a manageable mathematical framework.

Contribution

It introduces a reduction of the four-body problem to a single integral equation using separable potentials and symmetry considerations, applicable to both bosonic and fermionic systems.

Findings

Reduced four-body problem to a single integral equation

Utilized symmetry to simplify the eigenvalue problem

Applicable to zero-range interaction limits

Abstract

We consider the four-boson and 3+1 fermionic problems with a model Hamiltonian which encapsulates the mechanism of the Feshbach resonance involving the coherent coupling of two atoms in the open channel and a molecule in the closed channel. The model includes also the pair-wise direct interaction between atoms in the open channel and in the bosonic case, the direct molecule-molecule interaction in the closed channel. Interactions are modeled by separable potentials which makes it possible to reduce the four-body problem to the study of a single integral equation. We take advantage of the rotational symmetry and parity invariance of the Hamiltonian to reduce the general eigenvalue equation in each angular momentum sector to an integral equation for functions of three real variables only. A first application of this formalism in the zero-range limit is given elsewhere [Y. Castin, C. Mora, L. Pricoupenko, Phys. Rev. Lett. 105, 223201 (2010)].