  Perturbation theory is used systematically to investigate the symmetries of
the Dirac Hamiltonian and their breaking in atomic nuclei. Using the
perturbation corrections to the single-particle energies and wave functions,
the link between the single-particle states in realistic nuclei and their
counterparts in the symmetry limits is discussed. It is shown that the limit of
S-V=const and relativistic harmonic oscillator (RHO) potentials can be
connected to the actual Dirac Hamiltonian by the perturbation method, while the
limit of S+V=const cannot, where S and V are the scalar and vector potentials,
respectively. This indicates that the realistic system can be treated as a
perturbation of spin-symmetric Hamiltonians, and the energy splitting of the
pseudospin doublets can be regarded as a result of small perturbation around
the Hamiltonian with RHO potentials, where the pseudospin doublets are
quasidegenerate.
