induces a large effective chemical potential shift in the nanowire, which can push the topological phase transition to prohibitively large magnetic field strengths. To capture these features of the proximity effect, we consider an effective one-subband model to describe the system shown in Fig. \[fig:Fig1\], $H_\text{eff}=\int dx\,\Psi^\dagger(x)\mathcal{H}_\text{eff}\Psi(x)$, where the Hamiltonian density is given by ($\hbar=1$) $$\label{model}
\begin{aligned}
\mathcal{H}_\text{eff}&=\tau_3\biggl[-\partial_x\left(\frac{1}{2m(x)}\partial_x\right)-\mu(x)-\Delta_Z(x)\sigma_1\biggr] \\
    &+\frac{i}{2}[\alpha(x)\partial_x+\partial_x\alpha(x)]\sigma_3-\Delta(x)\tau_2\sigma_2,
\end{aligned}$$ $\Psi(x)=[\psi_\uparrow(x),\psi_\downarrow(x),\psi^\dagger_\uparrow(x),\psi^\dagger_\downarrow(x)]^T$ is a Nambu spinor of operators $\psi_\sigma^\dagger(x)$ \[$\psi_\sigma(x)$\] that create (destroy) a particle of spin $\sigma$ at position $x$ in the nanowire, while $\sigma_{1,2,3}$ ($\tau_{1,2,3}$) are Pauli matrices acting in spin (particle-hole) space. Note that the derivative operator in Eq.  acts on all quantities to its right, e.g., $\partial_x\alpha\Psi=(\partial_x\alpha)\Psi+\alpha(\partial_x\Psi)$, to ensure the Hermiticity of the Hamiltonian [@Reeg:2015; @Klinovaja:2015]. We assume that all parameters take different values in the normal ($-L<x<0$) and superconducting ($x>0$) regions of the nanowire [@step]; note that we take the superconducting region to be semi-infinite for analytical calculations ($L_s\to\infty$). In particular, due to renormalization by the proximity effect, we neglect the Rashba SOI $\alpha$ and Zeeman energy $\Delta_Z=g\mu_BB/2$ (arising from an external magnetic field $B$ applied parallel to the nanowire axis, with $g$ the $g$-factor and $\mu_B$ the Bohr magneton) in the superconducting region, such 