0 20 40 60 80 0.0 0.2 0.4 0.6 0.8 =10 ( =0.2) N=16 14 12 10 8 6 N=4 G/G$_{0}$ J/ 0 20 40 60 80 0.0 0.2 0.4 0.6 0.8 Fig. 4 N=16 14 12 10 8 6 N=4 G/G$_{0}$ = 5 ( =0.4) ω $_{l}$= ε $_{d}$- ε F J/ FIG. 3: Conductance G in units of quantum conductance G 0 = e $^{2}$/h plotted versus contact interaction J at resonance α = ε$_{F}$ for ξ = 10 and ξ = 5 computed using extended molecules with N = 4 , 6 , 8 , 10 , 12 , 14 , 16 sites (values increase upwards). energies Σ$_{x}$ instead of Σ 0 $_{,x}$, by devising certain diagrammatic approximations. We shall not pursue this line here, because as noted in Sect. II, the validity of such approximations is usually not transparent physically. We present instead a simple renormalization scheme, which turns out to work surprisingly well for the IRLM. Inspired by the rather general framework of the Landau Fermi liquid theory [54], we shall simply suppose that the aforementioned renormalization can be accounted for by means of a multiplicative ε -independent factor F , G $^{r}$( ε ) |$_{exact}$ ≃ F G $^{r}$( ε ). Then, the problem is to deduce this factor. For the IRLM at resonance, this can be done, because, as already noted in Sect. V A, the exact conductance is known, G |$_{α}$ = ε$_{F}$ = G 0 . This condition determines the needed factor at resonance for given values of N , ξ and J (remember that we keep β = 1), but F could dependent on α . Because G = G ( α ) reaches its maximum at α = ε$_{F}$ one can admit that, at least not too far away from resonance, F only slightly depends on α and neglect this dependence altogether. We have computed and examined the curves of G = G ( α ) obtained within this renor- malization procedure. The results are very encouraging. The renormalized curves are much less size-dependent than the unrenormalized ones. The smaller the values of ξ and J , the weaker is the dependence on N . For illustration, we present in Fig. 4 for comparison the renormalized curves along with those unrenormalized in the rather unrealistic situation of very large values of J = 10 and ξ = 10, which corresponds to a very large ratio J/ Γ = 50. The fact that, with increasing N , the renormalized curves tend to become size-independent much faster than the unrenormalized curves is clearly visible even in this less favorable case. 16