Integral equations for the four-body problem

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 (2010) 223201]. (C) 2011 Academic des sciences. Published by Elsevier Masson SAS. All rights reserved.