  We found the same staggering pattern in the evolution of the spin-orbit (SO)
and pseudospin-orbit (PSO) splittings in neutron drops obtained by the
energy-density functional (EDF) theory using SAMi-T and SLy5 parameterizations,
with and without the tensor terms respectively. This staggering evolution is
similar to the results recently obtained by the relativistic
Brueckner-Hartree-Fock (RBHF) theory with Bonn A potential, which was claimed
to be a tensor effect. In this work, we present an intuitive but essential
explanation on the origin of this staggering evolution in the Skyrme EDF
theory. We found that for both the SO and PSO partners, the energy splittings
mainly originate from the SO potential. The staggering evolution of both the SO
and PSO splittings is mainly due to the drastic change of density while
neutrons filling different single-particle orbits. The key to determine the
specific staggering pattern is the strength of the neutron density gradient
term and the SO density term in the SO potential, not necessarily the tensor
force.
