terms in the limit $\Delta_Z\gg\alpha/L$ [@inequality]. After expanding, we find a solvability condition given by $$\label{ABSB0}
\begin{aligned}
0&=4\Delta^2\left[2(v_F^2+\alpha^2)-(v_F^2-\alpha^2)\cos(2k_{so}L)e^{\chi_Z/2}\right]^2 \\
    &-E^2\bigl[16v_F^2\alpha^2+8v_F\alpha(v_F^2+\alpha^2)\chi_\Delta+(v_F^2-\alpha^2)^2\chi_\Delta^2\bigr]e^{\chi_Z} \\
    &+2\mu\Delta\chi_\Delta\sin(2k_{so}L)e^{\chi_Z/2}[2(v_F^4-\alpha^4) \\
    &-(v_F^2-\alpha^2)^2\cos(2k_{so}L)e^{\chi_Z/2}],
\end{aligned}$$ where $\chi_Z=2\Delta_ZL/\alpha$.

![\[analytics\] (a) The ABS energy, found by numerically solving Eq.  for $\Delta_Z=0.75\Delta_Z^c$, oscillates as $k_{so}L$ is varied. Points marked in the plot correspond to the values of $k_{so}L$ chosen in panels (b)–(d). (b) When tuning to the resonance condition $\cos(2k_{so}L)=0$, a subgap ABS energy is pinned exponentially close to zero as a function of Zeeman energy $\Delta_Z$. The dashed black curve corresponds to the asymptotic analytical expression given in Eq. . (c), (d) When $k_{so}L$ is tuned away from the resonance value, the ABS energy is no longer strictly pinned to zero. All plots were obtained for $\mu=0$, $v_F=10\alpha$, $L=604$ nm, $m=0.02m_e$, and $\Delta_Z^c/\Delta_0=7$. ](analytics.pdf){width="\linewidth"}

From Eq. , we see that $E=0$ is a valid solution for the ABS energy only for a single value of $\Delta_Z$, thus corresponding to a finely tuned zero-energy crossing. This behavior is very different than that of a semi-infinite topological superconducting nanowire, where a strict zero-energy MBS persists for all field strengths exceeding the topological phase transition.

While a strict $E=0$ solution is possible for only a 