content_image/442612.jpg content_image/442638.jpg content_image/442637.jpg content_image/442632.jpg content_image/442633.jpg content_image/442635.jpg content_image/442631.jpg content_image/442639.jpg content_image/442636.jpg content_image/442634.jpg
<img src='content_image/442612.jpg'>

From (22) we have 

<img src='content_image/442638.jpg'>

In order to have a full system of equations we add to (22) the definition of the chemical potential of the system 

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 

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 ). 

<img src='content_image/442637.jpg'>

<img src='content_image/442632.jpg'>

<img src='content_image/442633.jpg'>

By using these definitions we can prove the equations 

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 

<img src='content_image/442635.jpg'>

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 

<img src='content_image/442631.jpg'>

Here we have taken into account the equation (30) and that that Y$_{1}$ ( λ ) has not explicit dependence of λ . 

<img src='content_image/442639.jpg'>

<img src='content_image/442636.jpg'>

<img src='content_image/442634.jpg'>

