Superconductivity of the two-dimensional Hubbard model with a small U

Superconductivity of the two-dimensional Hubbard model with t' = 0 has been examined in the small U limit, where the matrix element of pair scattering is expressed as V-kk' = U+U(2)chi (k+k'). The susceptibility chi (k) of the unperturbed band has been evaluated accurately and the gap equation, which reduces to a seqular equation, has been solved precisely. The gap is found to be of the form alpha exp(-2t(2)/xU(2)), where x is the eigenvalue of the seqular equation. We have found the largest eigenvalue x is always positive (superconductive). The symmetry of the gap function is b(1g) for the electron density n(e) > 0.6 and b(2g) for n(e) < 0.6, depending on the peak position of <chi>(k). It is roughly ( pi, pi) for the former case and ( pi, 0) for the latter. The superconductivity seems to prevail even for n(e) --> 0. These results can be explained with a simple criterion.