external magnetic field. Then in section III we observe that those regions are stable against phase jumps in one-dimensional systems, while when the modulated wire is connected with an unmodulated wire, the locations of the localized modes are determined by which of the wires becomes TS. We also study the case with quasiperiodic modulation in the intersite hopping parameter, and find that multiple topological transitions into and out of TS states occur. Finally, in section IV, we summarize our findings.

Site level modulation by a single (quasi)periodic lattice potential
===================================================================

We consider a one-dimensional quantum wire parallel to the $\hat{{\mbox{\boldmath $x$}}}$ direction, coupled to a bulk superfluid whose surface is perpendicular to $\hat{{\mbox{\boldmath $z$}}}$. We study a tight-binding one-dimensional model of spin-$1/2$ fermions with the Rashba-type spin-orbit coupling, the mean-field coupling to the bulk superconductor, and the Zeeman energy due to the external magnetic field ${\mbox{\boldmath $B$}}$.

The Hamiltonian we have adopted is $$\begin{aligned}
\mathcal{H}
&=& -\frac{t}{2}\sum_{l=0}^{L-2}\sum_{\sigma=\uparrow,\downarrow}
(\hat c_{\sigma,l}^\dag \hat c_{\sigma,l+1} + \mathrm{h.c.})\nonumber\\
&+& \frac{\alpha}{2} \sum_{l=0}^{L-2} 
\left(
(\hat c_{\downarrow,l}^\dag \hat c_{\uparrow,l+1}
-\hat c_{\uparrow,l}^\dag \hat c_{\downarrow,l+1}) + \mathrm{h.c.}\right)\nonumber\\
&+& \sum_{l=0}^{L-1} \left(\Delta(\hat c_{\uparrow,l} \hat c_{\downarrow,l} + \mathrm{h.c.})
+\frac{2\Gamma}{\hbar}\hat{{\mbox{\boldmath $B$}}}\cdot{\mbox{\boldmath $S$}}_l\right)\nonumber\\
&+& \sum_{l=0}^{L-1} \sum_{\sigma=\uparrow,\downarrow}(t-\mu+\epsilon_{\sigma,l}) \hat n_{\sigma,l}
,
\label{eqn:Hamiltonian_gen}\end{aligned}$$ with $\Gamma \equiv \frac{g\mu_B}{2}B$. We 