that $\alpha(x)=\alpha\theta(-x)$ and $\Delta_Z(x)=\Delta_Z\theta(-x)$ \[$\theta(x)$ is a Heaviside step function\] [@neglect]. We neglect any orbital effects arising from the magnetic field [@olesia; @serra; @anton]. Pairing is induced only in the superconducting region of the nanowire, $\Delta(x)=\Delta\theta(x)$, while the mass is taken as $1/m(x)=(1/m)\theta(-x)+(1/m_s)\theta(x)$. The external magnetic field is incorporated in the superconducting region through the suppression of the bulk proximity gap, $\Delta=\Delta_0\sqrt{1-(B/B_c)^2}=\Delta_0\sqrt{1-(\Delta_Z/\Delta_Z^c)^2}$, where $B_c$ is the critical field at which superconductivity is destroyed (we additionally define the Zeeman energy induced in the quantum dot at the critical field as $\Delta_Z^c=g\mu_BB_c/2$). Finally, the chemical potential of the superconducting region is taken to be a large energy scale, $\mu(x)=\mu\theta(-x)+\mu_s\theta(x)$, where $\mu_s\gg\Delta$. We stress that our model \[Eq. \] does not have any topological phase regardless of the strength of the applied magnetic field due to the fact that $\alpha=0$ in the superconducting region.

The normal quantum dot ($-L<x<0$) is described by the Bogoliubov-de Gennes (BdG) equation $$\label{BdGw}
\left[\left(-\frac{\partial_x^2}{2m}-\mu\right)\tau_3+i\alpha\partial_x\sigma_3-\Delta_Z\tau_3\sigma_1\right]\psi_n(x)=E\psi_n(x).$$ To simplify the calculation, we assume that the spin-orbit energy $E_{so}=m\alpha^2/2$ is the largest energy scale in the normal region ($E_{so}\gg\Delta_Z,|\mu|$). A general solution to Eq.  to lowest order is then given by $$\label{solN}
\begin{aligned}
\psi_n(x)&=c_1\begin{pmatrix} v_{+} \\ -u_{+} \\ 0 \\ 0 \end{pmatrix}e^{-\Omega_{+}x/\alpha} 