retaining the same qualitative behavior obtained analytically.

This hierarchy of energy scales forms the central result of our paper, as it allows us to place bounds on the range of Zeeman fields over which a pinned ABS can be observed. It is also worth emphasizing that these bounds are consistent with experimental observations in epitaxial Al setups, where a zero-bias peak is typically observed for field strengths $\sim1$ T [@Deng:2016; @Zhang:2018; @Vaitiekenas:2018; @Deng:2018]. Assuming a spin-orbit energy for a typical semiconducting (InAs or InSb) nanowire of about $E_{so}\sim1$ meV (which corresponds to a spin-orbit strength $\alpha\sim0.8$ eV Å) and a $g$-factor $g\sim20$ [@Lutchyn:2018], and given that a typical quantum dot length is $L\sim150$ nm [@Deng:2018], the magnetic field strengths over which a pinned ABS can be observed are roughly given by 0.5 T $\lesssim B\lesssim$ 3 T, consistent with experimental observations.

Numerical Calculations {#numerical}
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In this section, we perform numerical simulations of the hybrid system presented in Fig. \[fig:Fig1\]. We find that even when we incorporate the superconductor explicitly, all of the qualitative features of our analytical model from Sec. \[analytical\] are reproduced in the numerical results. Specifically, we show explicitly how the system can be fine tuned to 