of Eq.  is positive definite (and $\ll1$), a solution to Eq.  always exists and there is always at least one spin-degenerate subgap ABS.

In the presence of a magnetic field, the spin-degenerate ABSs determined by Eq.  split, and it is possible to have an ABS near zero energy even in the limit of a short dot ($\alpha/L\gg\Delta$, or equivalently $\chi_\Delta\ll1$). We now determine the conditions necessary for an ABS to remain pinned near zero energy over a large range of magnetic field strength. From the form of the wave function in Eq. , we see that the Zeeman energy enters Eq.  at low energies ($E\to0$) through the factors $\exp(\pm\sqrt{\Delta_Z^2-\mu^2}L/\alpha)$. Thus, it is clear that these factors can lead to an ABS energy that decays exponentially with $\Delta_Z$ provided that the chemical potential is tuned within the Zeeman gap (such that $|\mu|<\Delta_Z$). Additionally, the Zeeman energy at which the ABS becomes pinned near zero energy is minimized by tuning to $\mu=0$, in which case the ABS energy is pinned for $\Delta_Z\gg\alpha/L$. To find under what conditions it is possible to have a pinned ABS, let us expand Eq.  to lowest order in the limits $E,|\mu|\ll\Delta,\Delta_Z,\alpha/L$ while keeping only the most relevant 