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Moment Approach for the 2D Attractive Hubbard Model

We constructed the one-particle spectral functions (diagonal and off-diagonal) which reproduce BCS for weak coupling and which take into account the effect of correlations on superconductivity in the attractive Hubbard model. The diagonal spectral function is composed of three peaks and the off-diagonal one is composed of two peaks. This ansatz satisfies the sum rules for the first six moments. Our solutions are valid for intermediate coupling, i.e., for $U/t \approx -4.0$. Our set of analytical equations for the unknown variables is self-consistent and has been solved numerically in lowest order of the order parameter. As a result, we obtain that the presence of the third band, or {\it upper Hubbard band}, strongly renormalizes the two lower bands, making that the energy gap be ${\bf k}$-dependent while the order parameter is pure s-wave. This shows that the order parameter and the gap are two different quantities.