〈 U$_{λ}$ ( β ) 〉 c 0 = − λ 4 4 1 4 4 3 3 2 2 1 + 3 1 3 2 1 2 − λ 3 3 1 2 2 1 − λ 2 2 + λ 4 2 1 3 4 2 + ... 1 1 2 2 3 3 4 4 + λ 4 8 1 2 3 4 FIG. 3: The first four orders of perturbation theory for 〈 U λ ( β ) 〉 c $_{0}$. 5 From (22) we have By using these definitions we can prove the equations In order to have a full system of equations we add to (22) the definition of the chemical potential of the system and as a result we obtain the stationary property: By using the definition (13) of the mass operator Σ$_{λ}$( k ) we can rewrite the functional Y$_{1}$ ( λ ) in the form where N$_{e}$ is the electron number. Equation (21) establishes the relation between thermo- dynamic potential and renormalized one-particle propa- gator. This last quantity depends on the auxiliary pa- rameter λ and (21) contains an additional integration over it and is awkward because that. We shall obtain more convenient equation for thermo- dynamic potential without such integration by λ . For that we shall introduce a special functional and prove the second form of stationary property if we take into account the Dyson equation for tunneling function T$_{λ}$ ( k ). Now we shall discuss the derivative by λ of the func- tional Y ( λ ). As was mentioned above there is twofold dependence of λ through Σ$_{λ}$( k ) and explicit through the factors λ n before skeleton diagrams of Y $^{′}$( λ ) Fig. 4. Due to the stationary property (30) we obtain where and Y $^{′}$( λ ) is constructed from skeleton diagrams without external lines and is depicted on Fig.4 The dependence on λ in the functional Y $^{′}$( λ ) is twofold, just through the dependence of the renormalized Green’s functions G$_{λ}$, T$_{λ}$, Λ$_{λ}$ and Σ$_{λ}$ and through the explicit fac- tors λ n in front of each diagram of Y $^{′}$( λ ). Here we have taken into account the equation (30) and that that Y$_{1}$ ( λ ) has not explicit dependence of λ . dY ( λ ) dλ = ∑ k δY ( λ ) δ Σ$_{λ}$( k ) δ Σ$_{λ}$( k ) δλ + ∂Y ( λ ) ∂λ |$_{Σ}$$_{λ}$ (31) = ∂Y $^{′}$( λ ) ∂λ |$_{Σ}$$_{λ}$ . Y$_{1}$ ( λ ) = − 1 β ∑ − → k ,σ,ω n [ln( ϵ ( − → k ) λ Λ$_{λ}$( k ) − 1) + T$_{λ}$ ( k ) λ Λ$_{λ}$( k )] exp( iω$_{n}$ 0 $^{+}$) , (26) Y ( λ ) = Y$_{1}$ ( λ ) + Y $^{′}$( λ ) , (25) δ Y ( λ ) δ Σ$_{λ}$( k ) = 0 , (30) Y$_{1}$ ( λ ) = − 1 β ∑ − → k $_{σ}$∑ ω n [ln( ϵ ( − → k )Σ$_{λ}$( k ) − 1) + T$_{λ}$ ( k )Σ$_{λ}$( k )] exp( iω$_{n}$ 0 $^{+}$) , (29) δβY ( λ ) δT$_{λ}$ ( k ) = 0 . (28) N$_{e}$ = ∑ − → k σ 1 β ∑ ω n G$_{σ}$ ( − → k | iω$_{n}$ ) exp( iω$_{n}$ 0 $^{+}$) , (24) λ F ( λ ) dλ = ∑ − → k σ 1 β ∑ ω n T$_{λ}$ ( k )Σ$_{λ}$( k ) exp( iω$_{n}$ 0 $^{+}$) = 1 β T r ( T$_{λ}$ Σ$_{λ}$) . (23) δβY$_{1}$ ( λ ) δT$_{λ}$ ( k ) = − λ Λ$_{λ}$( k ) = − Σ$_{λ}$( k ) , δβY $^{′}$( λ ) δT$_{λ}$ ( k ) = λ Λ$_{λ}$( k ) = Σ$_{λ}$( k ) , (27)