simplify the expressions, we will take $\hbar = k_\text{B} = 1$ in the following.

\[subsec:Fdu\_weak\] Solution near the second order transition
--------------------------------------------------------------

Here we solve the Gor’kov equations near the second order phase transition between the normal and superconducting states, namely assuming the limit $\Delta_0 \ll T_{c0}$.

To this purpose, it is useful to introduce the function $G_0$ which solves Eq. (\[Gorkov1\]) for $\Delta_0=0$, and also a closely related function $g_0$ which is the opposite of the kernel appearing in Eq. (\[Gorkov2\]). Those functions $G_0$ and $g_0$ are defined by the equations : $$\begin{aligned}
\left[\text{i}\,\omega_n + M(x) \right] G_0 &=& \delta(x-x^{\prime})\,\mathbb{1}\; ,\\
-\left[ \text{i}\,\omega_n - M^{\star}(x)\right] g_0 &=& \delta(x-x^{\prime})\,\mathbb{1}\, ,\end{aligned}$$ which are easily solved in momentum space, yielding : $$G_0 = \frac{1}{D_-}
\begin{pmatrix}
\text{i}\,\omega_n - \xi + \alpha\,p + h_z & h_x \\
h_x & \text{i}\,\omega_n - \xi - \alpha\,p - h_z
\end{pmatrix}\;,
\label{G0}$$ and: $$g_0 = -\frac{1}{D_+}
\begin{pmatrix}
\text{i}\,\omega_n + \xi + \alpha\,p - h_z & -h_x \\
-h_x & \text{i}\,\omega_n + \xi - \alpha\,p + h_z
\end{pmatrix}\;,
\label{g0}$$ where $\xi = \xi_p = \frac{p^2}{2\,m} - \mu$, and the $(p,\omega_n)$ arguments have been omitted in $G_0(p)=G_0(p,\omega_n)$ and $g_0(p)=g_0(p,\omega_n)$. The denominators $D_\pm$ are given by : $$D_{\pm} = \left[\text{i}\,\omega_n \pm 