single particle state running between RZ bands as $\phi_\mathrm{J}$ is changed. Increasing the number of phase jumps does not change the picture significantly, as long as the localization of the end modes is almost within a single section between phase jumps. This is observed in Fig. \[fig:phiJ\] (b); while the values of $\epsilon^+$ sometimes differ between systems with different numbers of phase jumps of $\pi$, the regions with $\epsilon^+\ll 1$ is not shifted or removed. Also we find in Fig. \[fig:phiJ\] (c–e) that the eigenvalues of $M_\mathrm{BdG}$ other than $\epsilon^\pm$ do not vanish in this case. There are only one pair of Majorana fermions appearing at the both ends of the system, rather than more than one pairs of them appearing also at phase jumps.

The results above reflect the fact that a phase jump does not change the topological character of the system. If both sides of the introduced phase jump are in the state characterized by the same topological quantum number, boundary states do not form at the phase jump.

Site level modulation limited to a part of the system
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![ (a) Value of $\epsilon^+$ plotted against $\mu$ for $n_\mathrm{B} = 100, 140, 180$ and $g=\sqrt5-2$, $\hat{{\mbox{\boldmath 