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We analyze solutions to a superconducting gap equation based on thetwo-dimensional Hubbard model with nearest and next-to-nearest neighborhopping. The Cooper pair potential can be calculated exactly and expressed interms of elliptic functions. The Fermi surfaces at finite temperature andchemical potential are calculated based on the exact two-body S-matrix of theHubbard model using the formalism we recently developed, which allows variationof hole doping. The resulting solutions to the gap equation are stronglyanisotropic, namely largest in the anti-nodal direction, and zero in the nodaldirections of the Brillouin zone. For U/t = 13 and t' /t =-0.3, appropriate toBSCO, and a physically natural choice for the cut-off, our self-containedanalytic calculations yield the gap in the anti-nodal direction Delta/t = 0.06and a maximum Tc/t = 0.04 at hole doping h=0.15. For phenomenological fits tothe Fermi surfaces for cuprates, we obtain the comparable value Tc/t = 0.03 atoptimal doping, both in good agreement with experiments. The superconductinggap is non-zero for all hole-doping h < 0.35, and increases all the way down tozero doping, suggesting that it evolves into the pseudogap.

Superconductivity in the two-dimensional Hubbard model based on the exact pair potential