set $t=1$ as the unit of energy and set $\hbar = 1$ in the following. In the case of $\hat{{\mbox{\boldmath $B$}}} = \hat{{\mbox{\boldmath $z$}}}$ the third line becomes $$\begin{aligned}
\sum_{l=0}^{L-1} \left(\Delta(\hat c_{\uparrow,l} \hat c_{\downarrow,l} + \mathrm{h.c.})
+\Gamma(\hat n_{\uparrow,l}-\hat n_{\downarrow,l})\right).\nonumber\end{aligned}$$ Here, $\hat c_{\sigma,l}$ annihilates a fermion with spin $\sigma(=\uparrow, \downarrow)$ at site $l (=0, 1, \ldots, L-1)$, $\hat n_{\sigma,l} \equiv \hat c^\dag_{\sigma,l} c_{\sigma,l}$, $t$ determines the nearest-neighbor hopping, $\alpha$ is the Rashba-type SOI, $\Delta$ is the coupling to the bulk superconductor, $\Gamma$ is the Zeeman energy, $\mu$ is the chemical potential, and $\epsilon_{\sigma,l}$ is the site energy for spin $\sigma$ on site $l$.

In this work we limit our discussion to the case with $\epsilon_{\sigma,l} = \epsilon_l$. In the following we introduce a quasiperiodic modulation to the site energy, and study the energy distribution of the single particle state and its correspondence with the realization of Majorana end modes.

![(Color online) The distribution of single particle state energy of eq. (\[eqn:Hamiltonian\_gen\]) for $L = 200$, $(\Gamma, \alpha)=(0.3, 0.3)$, $V_\mathrm{Q}=0, 0.2, 0.6, 1.25$, $\hat{{\mbox{\boldmath $B$}}} = \hat{{\mbox{\boldmath $z$}}}$. The single particle Hamiltonian, obtained by setting $\Delta = 0$ and $\mu=0$ in eq. (\[eqn:Hamiltonian\_gen\]), is diagonalized for $\kappa = 2\pi j/400$ ($j=0,1,.\ldots,400$) and 