+c_2\begin{pmatrix} u_{+} \\ -v_{+} \\ 0 \\ 0 \end{pmatrix}e^{\Omega_{+}x/\alpha}+c_3\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}e^{i[2k_{so}+(E+\mu)/\alpha]x}+c_4\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}e^{-i[2k_{so}+(E+\mu)/\alpha]x} \\
    &+c_5\begin{pmatrix} 0 \\ 0 \\ v_{-} \\ u_{-} \end{pmatrix}e^{-\Omega_{-}x/\alpha}+c_6\begin{pmatrix} 0 \\ 0 \\ u_{-} \\ v_{-} \end{pmatrix}e^{\Omega_{-}x/\alpha}+c_7\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}e^{-i[2k_{so}-(E-\mu)/\alpha]x}+c_8\begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}e^{i[2k_{so}-(E-\mu)/\alpha]x},
\end{aligned}$$ where $k_{so}=m\alpha$ is the spin-orbit momentum, $c_{1-8}$ are unknown coefficients to be found by imposing boundary conditions, $u_{\pm}=[\text{sgn}(E\pm\mu)/\sqrt{2}]\sqrt{1+ i\Omega_{\pm}/(E\pm\mu)}$, $v_{\pm}=(1/\sqrt{2})\sqrt{1- i\Omega_{\pm}/(E\pm\mu)}$, and $\Omega_{\pm}=\sqrt{\Delta_Z^2-(E\pm\mu)^2}$.

The superconducting region of the nanowire is described by the BdG equation $$\label{BdGs}
\left[\left(-\frac{\partial_x^2}{2m_s}-\mu_s\right)\tau_3-\Delta\tau_2\sigma_2\right]\psi_s(x)=E\psi_s(x).$$ Focusing only on subgap energies $E<\Delta$, the general solution to Eq.  that decays in the limit $x\to\infty$ is given by $$\label{wfs}
\begin{aligned}
\psi_s(x)&=e^{-\Omega_\Delta x/v_F}\left[c_9\begin{pmatrix} u_\Delta \\ 0 \\ 0 \\ v_\Delta \end{pmatrix}e^{ik_Fx}+c_{10}\begin{pmatrix} 0 \\ u_\Delta \\ -v_\Delta \\ 0 \end{pmatrix}e^{ik_Fx}+c_{11}\begin{pmatrix} 0 \\ -v_\Delta \\ u_\Delta \\ 0 \end{pmatrix}e^{-ik_Fx}+c_{12}\begin{pmatrix} v_\Delta \\ 0 \\ 0 \\ u_\Delta \end{pmatrix}e^{-ik_Fx}\right],
\end{aligned}$$

where $k_F=\sqrt{2m_s\mu_s}$ is the Fermi momentum, $v_F=\sqrt{2\mu_s/m_s}$ is the Fermi velocity, $u_\Delta(v_\Delta)=(1/\sqrt{2})\sqrt{1\pm i\Omega_\Delta/E}$, and $\Omega_\Delta=\sqrt{\Delta^2-E^2}$. Although we focus on a model in which topological superconductivity is not possible (as we have fixed $\alpha=\Delta_Z=0$ in the superconducting region), we note that Eq.  is valid to leading order 