Separable effective interaction for the trapped Fermi gas: The BEC-BCS crossover

We extend a recently introduced separable interaction for the unitary trapped Fermi gas to all values of the scattering length. We derive closed expressions for the interaction matrix elements and the two-particle eigenvectors. At unitarity we analytically demonstrate the convergence of this interaction to the zero-range two-body pseudopotential for s-wave scattering. We apply this effective interaction to the three- and four-particle systems along the BEC-BCS crossover and find that their low-lying energies exhibit convergence in the regularization parameter that is much faster than for the conventional renormalized contact interaction. We find similar convergence properties of the three-particle free energy at unitarity.