8 FIG. 4. Vertex correction to the conductivity for the electron- hole symmetric case. FIG. 6. (Color online) Drude conductivity vs. density of states for filling n = 0 . 7. FIG. 7. Vertex correction to the conductivity for filling n = 0 . 7. FIG. 5. Ratio r = | Δ σ | /σ 0 for the electron-hole symmetric case. and where we subtract the Fermi energy and the self-energy from their values in the electron-hole symmetric case n$_{f}$ = 1 / 2 , n$_{c}$ = 1 / 2. Explicit formulas are more involved and we do not present them here. Second, as discussed in Ref. 22, a nontrivial solution at zero temperature ex- ists only for 1 / 2 < n < 3 / 2. Outside this region the f -electron energy level is either empty ( n ≤ 1 / 2) or fully filled ( n ≥ 3 / 2). We plotted the Drude conductivity and DOS in Fig. 6 for a total filling n = 0 . 7. The density of states at the Fermi energy is almost constant within the interval 0 ≤ U ≤ 0 . 82, within which the extended electrons scatter on f -electrons. The extended electrons go over to a Fermi gas in both limiting values of inter- action, hence the Drude conductivity diverges at both ends. The asymptotics is, however, different at the two ends. The asymmetry between the two limiting interac- tion strengths is demonstrated in Fig. 7, where we plot- ted the vertex correction to the Drude conductivity. It is two orders smaller than the mean-field one. This is clearly seen from their ratio plotted in Fig. 8. The ratio is no longer a monotonically increasing function as in the electron-hole symmetric case, but reaches maximum at U$_{m}$ ≈ 0 . 48. We studied effects of elastic scatterings of mobile elec- trons on either thermally equilibrated or frozen, ran- VII. CONCLUSIONS -0.018 -0.016 -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002  0  0  0.2  0.4  0.6  0.8  1 U  0  2  4  6  8  10  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8 U 0 F  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  0.2  0.4  0.6  0.8  1 r U -0.01 -0.0098 -0.0096 -0.0094 -0.0092 -0.009 -0.0088  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8 U y = ∫ ∞ −∞ dϵρ ( ϵ ) ϵ ( E$_{F}$ − ℜ Σ − ϵ ) 2 + ℑ Σ 2 +