Gutzwiller variational theory for the Hubbard model with attractive   interaction
  We investigate the electronic and superconducting properties of a negative-U Hubbard model. For this purpose we evaluate a recently introduced variational theory based on Gutzwiller-correlated BCS wave functions. We find significant differences between our approach and standard BCS theory, especially for the superconducting gap. For small values of $|U|$, we derive analytical expressions for the order parameter and the superconducting gap which we compare to exact results from perturbation theory. 
Joerg Buenemann, Florian Gebhard, Katalin Radnoczi, Patrik Fazekas