($N_\downarrow + N_\uparrow < L$). Remarkably, even when the chemical potential lies in one of such bands, a TS state with end Majorana fermions can be formed.

However, as $\Delta$ is increased, smaller features in the single particle state distribution, which remained in the structure of the value of $\epsilon^+$ for smaller $\Delta$, gradually disappear and only the widest single-band regions remain visible. Similar simplification of the phase diagram is also observed in Fig. 2 of Ref.  for a quasiperiodic system of spinless fermions. Finally, for $\Delta \gtrsim \Gamma$, the in-gap state disappears and the eigenvalue spectrum of $M_\mathrm{BdG}$ has a gap of the order of $\Delta$, regardless of the value of $V_\mathrm{Q}$, so that the system is topologically trivial.

In summary, the new topologically non-trivial regions with end Majorana fermions appear for general site energy modulations of the type of eq. (\[eqn:sitePotential\]), and their existence is not limited to some special irrational values of $g=\kappa/(2\pi)$. The commensurate case can be considered as a kind of multi-band wire. The possibility of TS states with end Majorana modes has also been studied in multi-band systems.[@Potter2010; @Lutchyn2011a; @Rieder2012] We have observed that while the range of the chemical potential reflects the single 