Anomalous superconductivity in coupled Luttinger liquids

According to Anderson ((A], pag. 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 hamil-tonian. 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 nd that the Luttinger interaction enhances strongly T c if the intrachain Luttinger interaction is much bigger than the interchain interaction.