  The pseudospin symmetry (PSS) has been studied extensively for bound states.
Recently we justified rigorously that the PSS in single particle resonant
states is exactly conserved when the attractive scalar and repulsive vector
potentials of the Dirac Hamiltonian have the same magnitude but opposite sign
[PRL 109, 072501 (2012)]. To understand more deeply the PSS, we focus on
several issues related to the exact conservation and breaking mechanism of the
PSS in single particle resonances. In particular, we are interested in how the
energy and width splittings of PS partners depend on the depth of the scalar
and vector potentials. We investigate the asymptotic behaviors of radial Dirac
wave functions. Spherical square well potentials are employed in which the PSS
breaking part in the Jost function can be well isolated. By examining the zeros
of Jost functions corresponding to small components of the radial Dirac wave
functions, general properties of the PSS are analyzed. By examining the Jost
function, the occurrence of intruder orbitals is explained and it is possible
to trace continuously the PSS partners from the PSS limit to the case with a
finite potential depth. The dependence of the PSS in resonances as well as in
bound states on the potential depth is investigated systematically. We find a
threshold effect in the energy splitting and an anomaly in the width splitting
of pseudospin partners when the depth of the single particle potential varies
from zero to a finite value. The conservation and the breaking of the PSS in
resonant states and bound states share some similar properties. The appearance
of intruder states can be explained by examining the zeros of Jost functions.
Origins of the threshold effect in the energy splitting and the anomaly in the
width splitting of PS partners, together with many other problems, are still
open and should be further investigated.
