single value of $\Delta_Z$, it is possible for a trivial ABS to persist near zero energy over a wide range of magnetic field strength. From Eq. , we see that this behavior is most pronounced in the presence of Fermi velocity mismatch ($v_F\neq \alpha$) at $\mu=0$ when $\cos(2k_{so}L)=0$, corresponding to a finely tuned resonance condition determined solely by the spin-orbit length $1/k_{so}$ and the length of the quantum dot $L$. When the resonance condition is satisfied, the ABS energy is given by $$\label{EABS}
E=(2\alpha/L)e^{-\Delta_ZL/\alpha},$$ where we have additionally assumed that $v_F\gg\alpha$ \[and $(v_F/\alpha)\chi_\Delta\gg1$\], though we note that this assumption changes only the prefactors in Eq.  and does not affect the exponential behavior [@nomismatch].

If the system is slightly off-resonance with $\cos(2k_{so}L)\neq0$, a constant shift of the ABS energy is induced, given by $(\alpha/L)|\cos(2k_{so}L)|$, which remains small as long as $|\cos(2k_{so}L)|\ll\chi_\Delta$. If the system is on resonance $\cos(2k_{so}L)=0$ but is tuned away from $\mu=0$, this also induces a constant shift of the ABS energy given by $|\mu|/2$. While we consider in this section only small deviations away from $\mu=0$, we show numerically in Appendix \[appA\] that the pinning behavior can withstand larger deviations provided that one simultaneously alters the resonance condition. 