\right )},\nonumber\\
\mathcal{A}={\left( \begin{array}{cc}
0 & -{M^\ast}'\\
{M^\ast}' & 0
\end{array}
\right )}&,\quad
\mathcal{B}={\left( \begin{array}{cc}
0 & \frac{{M^\ast}'}{{M^\ast}^2}\\
-\frac{{M^\ast}'}{{M^\ast}^2} & 0
\end{array}
\right )}.\end{aligned}$$

According to Eq. (\[FWH\]), the Dirac Hamiltonian is transformed by the reconstituted FW transformation as $$\label{eq:HFW}
{\left( \begin{array}{cc}
\mathcal{H}^{\rm (F)}_{\rm rFW} + M & O(\frac{1}{{M^\ast}^4}) \\
O(\frac{1}{{M^\ast}^4}) & \mathcal{H}^{\rm (D)}_{\rm rFW} - M
\end{array}
\right)}.$$

Hereafter, we will focus on the single-particle states in the Fermi sea, which correspond to their counterparts in the non-relativistic framework. Therefore, $\mathcal{H}^{\rm (F)}_{\rm rFW}$ will be investigated in detail, and its superscript will be omitted when there is no confusion.

The expansion of $\mathcal{H}_{\rm rFW}$ up to the $1/{M^\ast}^3$ order is carefully worked out as

\[neq\] $$\begin{aligned}
\mathcal{H}_{\rm rFW,0}=\,&\Sigma,\label{neq1}\\
\mathcal{H}_{\rm rFW,1}=\,&-\frac{\textrm{d}}{\textrm{d}r}\frac{1}{2M^\ast}\frac{\textrm{d}}{\textrm{d}r}+\frac{\kappa(\kappa+1)}{2M^\ast r^2},\label{neq2}\\
\mathcal{H}_{\rm rFW,2}=\,&-\frac{\Delta'}{4{M^\ast}^2}\frac{\kappa}{r}+\frac{\Sigma''}{8{M^\ast}^2},\label{neq3}\\
\mathcal{H}_{\rm rFW,3}=\,&-p^2\frac{1}{8{M^\ast}^3}p^2 +\frac{\Sigma'^2}{8{M^\ast}^3}-\frac{3S'\Sigma'}{8{M^\ast}^3},\label{neq4}\end{aligned}$$

where $$p^2=-\frac{\textrm{d}^2}{\textrm{d}r^2}+\frac{\kappa(\kappa+1)}{r^2}.$$

It is noted that operators with higher-order derivatives appear from $\mathcal{H}_{\rm rFW,3}$, and thus the eigenequation containing up to the second derivatives reads $$\label{eq:Sch2}
  \left[\mathcal{H}_{\rm rFW,0} + \mathcal{H}_{\rm rFW,1} + \mathcal{H}_{\rm rFW,2}\right] \varphi_k(r) = \varepsilon_k\varphi_k(r),$$ with the normalization condition $\int \mathrm{d}r\,\varphi^2(r) = 1$. Following the same way in Ref. [@Guo2019Phys.Rev.C99.054324], the eigenequation (\[eq:Sch2\]) will be solved and the higher-order term $\mathcal{H}_{\rm rFW,3}$ will be calculated by the perturbation theory in the 