〈 U ( β ) 〉 c 0 = 1 2 3 4 3 4 − λ 4 + λ 4 2 1 2 3 4 1 2 5 6 − λ 6 8 1 2 3 5 6 4 − λ 6 1 2 3 4 5 6 + 3 λ 6 2 − λ 8 1 2 3 4 5 6 7 8 − + 1 2 3 4 5 6 7 8 + λ 8 6 1 2 3 4 5 6 7 8 + λ 8 1 2 3 4 5 6 7 8 +2 λ 8 1 2 3 4 5 6 7 8 − λ 8 − − λ 8 48 1 2 3 4 5 6 7 8 $^{+}$... + λ 8 2 1 2 3 4 5 6 7 8 V ∫ 0 dλ λ { − λ 2 1 2 } FIG. 7: Vacuum diagrams of the system in superconducting state after integrating by interaction constant. ample of vacuum diagram of eighth order of perturbation theory is presented on the Fig. 9. Only all the three pos- sibilities give us the correct coefficient − 1 8 − 1 4 − 1 8 = − 1 2 of diagram. Other possibilities don’t exist. These examples demonstrate the general statement that the integrand of the evolution operator average can be presented itself as a product of λ $^{2}$G $^{c}$Λ of the form 〈 U ( β ) 〉 c 0 = − V ∫ 0 dλ λ ∑ αα ′ ∑ xx ′ αα $^{′}$λ $^{2}$G c αα ′ ( x | x $^{′}$| λ )Λ α $^{′}$α ( x $^{′}$| x | λ ) = (17) = − V ∫ 0 dλ λ ∑ αα ′ ∑ xx ′ G c αα ′ ( x | x $^{′}$| λ )Σ α $^{′}$α ( x $^{′}$| x | λ ) = − V ∫ 0 dλ λ T r ( ˆ G $^{c}$( λ ) ˆ Σ $^{c}$( λ )) . This expression for renormalized thermodynamic poten- tial of the strongly correlated system contains additional integration over the integration strength λ and is awk- ward because it. Equation (18) generalizes the result of Luttinger and Ward [24] proved for non-correlated many- electron system in normal state. Our generalization has been obtained for the case of strong correlations of special kind which contains one un- correlated subsystem and one strongly correlated and we admit also the existence of superconductivity in both of where the operators ˆ G $^{c}$( λ ) and ˆ Σ $^{c}$( λ ) have the matrix elements G c αα ′ ( x | x $^{′}$| λ ) and Σ αα $^{′}$( x | x $^{′}$| λ ) correspondingly. Index λ underline that these quantities depend of the auxiliary constant of integration λ . Therefore the thermodynamic potential of our system F is equal to F = F$_{0}$ + 1 β V ∫ 0 dλ λ T r [ ˆ G $^{c}$( λ ) ˆ Σ $^{c}$( λ )] . (18) 7