we can further explain the various characteristics of the spectrum around the Fermi energy ($\epsilon=0$) reported in Ref. [@Hotta]. For instance, Eq. (\[SSDspectrum\]) for a fixed $\beta(=N-m)$ leads to $$\epsilon_L = -\frac{C}{L^2},
\label{FSS}$$ with $C := \pi^2\beta(\beta-1)$, which is in good agreement with the size dependence found in Ref. [@Hotta]. Note that this scaling relation contains no higher order correction of $O(L^{-3})$. Also from Eq. (\[SSDspectrum\]), we obtain the density of state (DOS) around the Fermi level as $$D(\epsilon) = \left|\frac{d \beta}{d\epsilon}\right| =\left(\frac{L}{\pi}\right)^2\frac{1}{2\beta-1}\simeq \frac{L}{2\pi} \frac{1}{\sqrt{|\epsilon|}}$$ for $|\epsilon|  \gg 1/L^2$. For a sufficiently long chain, the DOS diverges with $|\epsilon|^{-1/2}$, which is also consistent with the numerical result in Ref. [@Hotta]. For $|\epsilon| \ll 1/L^2$, however, $D \simeq (\frac{L}{\pi})^{2}$(=const), which is a cutoff bound in the vicinity of the Fermi level.

![Comparison of single-particle wavefunctions between the lattice and continuous SSD systems. The chain length and the chemical potential are set to $L=101$ and $\mu=0$, respectively. In the corresponding continuous SSD, the total number of fermions is fixed as $N=50$. The lattice wavefunctions with the geometric correction ($\psi^{\rm ssd}_m (n)$) are plotted as open circles for $m=0, 1, 2, 25$, while the continuum wavefunctions ($\psi^{m}_{N-m}(x)$) are shown as solid lines. Note 