$a_0= 0.6$ fm, which are the typical values for neutrons in the nucleus $^{208}\textrm{Pb}$. The mass of nucleon is taken as $M=939.0$ MeV.

The Dirac equation (\[eq:Dirac1\]) is solved in coordinate space by the shooting method [@Meng1998Nucl.Phys.A635.3--42] within a spherical box with radius $R_{\rm box}= 20$ fm and mesh size $\textrm{d}r = 0.05$ fm. The single-particle energies and densities thus obtained will serve as benchmarks, labelled as Exact in the tables and figures. The non-relativistic equation (\[eq:Sch2\]) is also solved in coordinate space by the shooting method with the same box and mesh sizes.

Single-particle spectrum {#sec:IIIA}
------------------------

![(Color online) (a) Discrepancy between the neutron single-particle energies obtained by the reconstituted FW transformation and the exact ones as a function of the single-particle energy. The squares, circles, and triangles represent the energy discrepancy at the $1/M^\ast$, $1/{M^\ast}^2$, and $1/{M^\ast}^3$ orders for the reconstituted FW transformation, respectively. (b) Discrepancy at the $1/{M^\ast}^3$ order on a smaller scale.[]{data-label="fig1"}](Fig1.eps){width="45.00000%"}

In Fig. \[fig1\], the discrepancy between the neutron single-particle energies obtained by the reconstituted FW transformation and the exact ones is shown as a function of the single-particle energy. In the present case of $^{208}$Pb, the last occupied orbital is the $3p_{1/2}$ state, whose 