standard forms :

$\sigma_x = \begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}$; $\sigma_y = \begin{pmatrix}
0 & -\text{i} \\ \text{i} & 0
\end{pmatrix}$; $\sigma_z = \begin{pmatrix}
1 & 0 \\ 0 & -1
\end{pmatrix}$.

The kinetic term $\mathcal{H}_0$ corresponds to a purely quadratic dispersion with an effective mass $m$ and a chemical potential $\mu$. The Zeeman Hamiltonian $\mathcal{H}_{\vec{h}}$ incorporates the effect of a tilted field with respect to the wire direction $x$. The Rashba Hamiltonian $\mathcal{H}_\text{SO}$ originates from the mirror symmetry breaking due to the substrate, and its strength is given by the spin-orbit velocity $\alpha$. Finally, $\mathcal{H}_\text{BCS}$ encodes the standard s-wave superconducting correlations within the BCS model.

Because of the Zeeman splitting, the singlet superconducting order parameter can oscillate along the $x$-axis as: $$\Delta(x) = \Delta_0\,e^{\text{i} q x}\;,
\label{Delta}$$ where $\Delta_0$ is independent of the position $x$, and $q$ is the wave vector of the inhomogeneous order parameter.

\[subsec:Gorkov\_eq\] Gor’kov equations
---------------------------------------

The Gor’kov Green’s functions are defined as the following $2$-point correlators between the field operators: $$\begin{aligned}
G_{\alpha \beta}(x,x',\tau) &=& - \left\langle T_{\tau}\,\psi_{\alpha}(x,\tau)\,\psi^{\dagger}_{\beta}(x',0)\right\rangle \;,\label{G}\\
F^{\dagger}_{\alpha \beta}(x,x',\tau) &=& \left\langle T_{\tau}\,\psi^{\dagger}_{\alpha}(x,\tau)\,\psi^{\dagger}_{\beta}(x',0)\right\rangle \;,\label{Fdag}\end{aligned}$$ where $T_\tau$ is the time-ordering product for the imaginary time $\tau$, and the brackets denote averaging 