6 FIG. 4: The skeleton diagrams for functional Y $^{′}$( λ ). The rectangles depict the irreducible Green’s functions. The double dashed lines depict the tunneling renormalized Green’s functions T αα $^{′}$( x | x $^{′}$). since for λ = 0 the perturbation is absent Y ( λ = 0) = 0 and F ( λ = 0) = F$_{0}$. Now we set λ = 1 and obtain where the usual convention about summation over the repeated indices has been adopted. As a result we obtain the stationarity property (29) of the functional Y ( λ ) versus the change of the mass oper- ator Σ$_{λ}$. This mass operator for λ = 1 coincides with correlation function of our strongly correlated model. Now it is necessary to find a relation between the ther- modynamic potential F ( λ ) and the functional Y ( λ ). Consider first the value of the derivative dY ( λ ) dλ . The λ dependence of the functional Y ( λ ) is of two kinds: through Σ$_{λ}$ and also explicit through the factors λ n in front of the skeleton diagrams for the functional Y $^{′}$( λ ). Due the stationarity property (29) we obtain We have further developed the diagrammatic theory proposed for strongly correlated systems many years ago to establish the stationarity property of the thermody- namic potential in the superconducting state of the Hub- bard model. with the stationarity property Here we took into account that the Y$_{1}$ ( λ ) part of func- tional Y ( λ ) (see equations (21) and (28)) does not ex- plicitly dependent on λ . By using the definitions of Y $^{′}$( λ ) (see Fig.4) and of Λ$_{λ}$ (see Fig. 2) it is easy to establish the property: First, we have introduced the notion of charge quan- tum number which gives the possibility to consider the presence of irreducible Green’s functions with an arbi- trary number of creation or annihilation Fermi- operators in superconducting state. We have introduced the notion of tunneling Green’s function and its mass operator, which turns out to be equal to the correlation function of the fermion system. We have proven the existence of the Dyson equation for this function and establish the exact relation between the thermodynamic potential and renormalized one-particle propagator. This relation contains an additional integra- tion over the auxiliary constant of interaction λ . We have constructed a special functional based on the skeleton diagrams for the propagator and for the evolu- tion operator which contain the irreducible Green’s func- tions and full tunneling Green’s functions. We have proven the existence of the stationarity prop- erty of this functional and establish its relation with ther- modynamic potential. From the equations (20) and (35) we have and we therefore obtain F = F$_{0}$ + Y (1) , (38) δF δ ̂ Σ = 0 . (39) dY ( λ ) dλ = δY ( λ ) δ Σ$_{λ}$ d Σ$_{λ}$ dλ + ∂Y ( λ ) ∂λ |$_{Σ}$$_{λ}$ (34) = dY ( λ ) dλ |$_{Σ}$$_{λ}$ = dY $^{′}$( λ ) dλ |$_{Σ}$$_{λ}$ . λ dY ( λ ) dλ = λ ∂Y $^{′}$( λ ) ∂λ |$_{Σ}$$_{λ}$ = 1 2 β T r ( λ ̂ T$_{λ}$ ̂ Λ$_{λ}$) = 1 2 β T r ( ̂ T$_{λ}$ ̂ Σ$_{λ}$) . (35) λ dY ( λ ) dλ = 1 2 β T r ( ̂ T$_{λ}$ ̂ Σ$_{λ}$) = λ dF ( λ ) dλ , (36) F ( λ ) = Y ( λ ) + F$_{0}$, (37) IV. CONCLUSIONS Y $^{′}$( λ ) = 1 $_{2}${ λ + λ 4 24 + λ 2 4 − ... + } G (0) + λ 4 192 λ 3 24 −