content_image/427248.jpg content_image/427252.jpg content_image/427257.jpg
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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 

<img src='content_image/427252.jpg'>

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 

<img src='content_image/427257.jpg'>

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 

