in the limit $\mu_s\gg E_{so,s},\Delta_{Z,s}$ even if $\alpha_s\neq0$ and $\Delta_{Z,s}\neq0$ in the superconducting region. Therefore, the following calculation applies to any general case for which $\mu_s\gg E_{so,s},\Delta_{Z,s}$. We show in Appendix \[appA\] that similar results to those presented in this section can also be found assuming the SOI and Zeeman energy to be spatially uniform, emphasizing that the parameter renormalization caused by the proximity effect does not play a crucial role.

To determine the spectrum of ABSs, we must impose three boundary conditions [@compose; @Peter]. First, we impose that the wave function vanishes at the free end of the system, $\psi_n(-L)=0$. Second, the wave function should be continuous at $x=0$, $\psi_n(0)=\psi_s(0)$. And third, the quasiparticle current should be conserved, leading to the condition $\hat v_n\psi_n(0)=\hat v_s\psi_s(0)$, where $\hat v_n=(-i\partial_x/m)\tau_3-\alpha\sigma_3$ and $\hat v_s=(-i\partial_x/m_s)\tau_3$ are the velocity operators \[note that this boundary condition can also be obtained by directly integrating Eq. \]. These three boundary conditions can be arranged as a single equation of the form $M\vec c=0$, where $\vec c$ is a vector of unknown coefficients and $M$ is a matrix (which is too cumbersome to spell out here explicitly). This matrix equation has a nontrivial solution if $$\label{solvability}
\det(M)=0,$$ and 