over the thermal equilibrium distribution at temperature $T$ and chemical potential $\mu$. The electronic field operators obeys the fermionic anticommutation relations at equal (imaginary) times.

We follow the standard procedure [@abrikosov_methods_1965; @kopnin_theory_2001] to establish the differential equations for the Gor’kov Green’s functions. First, the equations of motion for the field operators $\psi_{\alpha}(x,\tau)$ and $\psi_{\alpha}^\dagger(x,\tau)$ are obtained by evaluating their commutators with the Hamiltonian Eq. (\[H\]). Second, multiplying the latter equations of motion by the field operators at $(x',0)$ and averaging other the thermal distribution leads to relations between the correlators Eqs. (\[G\],\[Fdag\]) and their $\partial_\tau$ derivatives with respect to imaginary time. Due to invariance with respect to time translations in $\tau$, it is then convenient to express those equations in Fourier space of Matsubara frequencies $\omega_n$ :

( \_n + M(x))G + (x) \_yF\^ &=& (x-x\^) ,\
( \_n - M\^(x))F\^ - \^(x)\_yG &=& 0 ,

where $G =G(x,x',\omega_n) $ and $F =F(x,x',\omega_n) $ are two by two matrices defined in spin space, and $\omega_n = 2\,\pi\,k_\text{B}\,T\,\left(n + 1/2 \right)$, $n$ being integer. The differential operator $M(x)$ is defined by : $$M(x) =  \frac{\hbar^2}{2\,m}\,\partial^2_x + \mu - \vec{h}\cdot\vec{\sigma} - \alpha\,\sigma_z\,\frac{\hbar}{\text{i}}\,\partial_x \; ,$$ where $\mathbb{1}$ corresponds to the identity matrix. To 