Symmetries and Supersymmetries of the Dirac Hamiltonian with Axially Deformed Scalar and Vector Potentials

We consider several classes of symmetries of the Dirac Hamiltonian in $3+1$ dimensions, with axially deformed scalar and vector potentials. The symmetries include the known pseudospin and spin limits and additional symmetries which occur when the potentials depend on different variables. Supersymmetries are observed within each class and the corresponding charges are identified.

Figure 1

Typical spectra of good and broken SUSY. The operators $Q_{-}$ and $Q_{+}$ connect degenerate states in the $H_{+}$ and $H_{-}$ sectors.

Figure 2

Grouping of deformed shell-model states $[N=4,n_3,\Lambda ]\Omega$, exhibiting a pattern of (a) good SUSY, relevant to the pseudospin symmetry limit, and (b) broken SUSY, relevant to the spin symmetry limit. $N$ and $n_3$ are harmonic oscillator quantum numbers. $\Lambda$ ($\tilde{\Lambda }$) is the orbital (pseudo-orbital) angular momentum projection along the symmetry $z$ axis.