The Two - Dimensional Attractive Hubbard Model: Highly Non-Linear Superconductivity With Sum Rules

We use the moment approach of Nolting (exact sum rules) (Z. Physik 255, 25 (1972)) for the attractive Hubbard model in the superconducting phase. Our diagonal and off - diagonal spectral functions are constructed and evaluated with the sum rules. They reduce to the $BCS$ limit for weak interaction. However, the presence of correlations modify the $BCS$ picture dramatically. For example, due to the presence of correlations we have postulated a three - pole ansatz for the diagonal Green function, $G(\vec{k},\omega)$, while the off - diagonal one, $B(\vec{k},\omega)$, is supposed to have two poles. In the paper we present results for the three spectral weights of the diagonal Green function, $\alpha_j(\vec{k})$, j = 1,2,3. Our results compare reasonably well with more elaborated auto - consistent highly non - linear equations (double fluctuation calculations in the $T-$ Matrix approach of one of the authors). Then, the physical picture which emerges is that the lower Hubbard band is split due to the superconducting gap and the upper Hubbard band remains mostly unmodified.