Anomalous superconductivity in coupled Luttinger liquids

According to Anderson ([1], p. 54) "the equivalent of BCS theory for a fully spin-charge separated Luttinger liquid has not been formally worked out". We consider a model for two one-dimensional spinning Luttinger liquids coupled via a Cooper tunnelling Hamiltonian. We show that the partition function is the four-dimensional integral of an exponential whose exponent has an extremal point obtained solving an anomalous non-BCS self-consistence equation. If the extremal point is a global minimum the model is completely solved by the saddle point theorem and the anomalous gap generation is proved. We find that the Luttinger interaction enhances strongly Tc if the intrachain Luttinger interaction is much bigger than the interchain interaction.