on the residual Fermi surface through scattering in the Cooper channel. Earlier numerical renormalization group studies on the 2-dimensional Hubbard model were interpreted as pointing towards a similar pairing mechanism. 19,20 A key feature of the present model is the presence of a finite energy Cooperon resonance in the pseudogap which is generated in association with the partial truncation of the Fermi surface. D-wave pairing follows on the remnant Fermi surface through the coupling to the Cooperon. The properties of a single 4-leg Hubbard ladder with open boundaries have been studied extensively in both the weak and strong coupling limits. We consider here the former with equal nearest neighbor hopping t$_{0}$ , along the legs and rungs. In this case the 4 bands split into two band pairs. The inner pair, A$_{1}$$_{,}$$_{2}$ , are standing waves on the rungs with wavevectors (2 π/ 5 , 3 π/ 5). At half filling the corresponding Fermi wavevectors are k$_{FA}$$_{1}$ = ± 3 π/ 5 and k$_{FA}$$_{2}$ = ± 2 π/ 5 leading to a common Fermi velocity, v$_{FA}$ = 2 t$_{0}$ sin(2 π/ 5). The outer band pair, B$_{1}$$_{,}$$_{2}$ , have Fermi wavevectors K$_{FB}$$_{2}$ = ± π/ 5 and K$_{FB}$$_{1}$ = ± 4 π/ 5 and a smaller Fermi velocity, v$_{FB}$ = 2 t$_{0}$ sin( π/ 5). We obtain a band structure of four bands with energies where ϵ$_{‖}$ ( k ) represents the dispersion along the ladder. The annihilation (creation) operators of electrons of the outer and inner bands, denoted as B$_{1}$$_{,}$$_{2}$, B † 1 , 2 and A$_{1}$$_{,}$$_{2}$, A † 1 , 2 respectively, are 3 B$_{1}$ = 4 ∑ n =1 sin( πn/ 5) c$_{n}$ ; B$_{2}$ = 4 ∑ n =1 sin(4 πn/ 5) c$_{n}$ ; A$_{1}$ = 4 ∑ n =1 sin(2 πn/ 5) c$_{n}$ ; A$_{2}$ = 4 ∑ n =1 sin(3 πn/ 5) c$_{n}$, (2.2) E$_{A}$$_{1}$$_{,}$$_{2}$ ( k ) = ϵ$_{‖}$ ( k ) ∓ 2 t$_{0}$ cos(2 π/ 5) , E$_{B}$$_{1}$$_{,}$$_{2}$ ( k ) = ϵ$_{‖}$ ( k ) ∓ 2 t$_{0}$ cos( π/ 5) , (2.1) II. FOUR LEG HUBBARD LADDERS