Pseudospin symmetry in single-particle resonances in spherical square wells

Background: 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 [Phys. Rev. Lett. 109, 072501 (2012)].

Purpose: To understand more deeply the PSS in single-particle resonant states, 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.

Methods: 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.

Results: By examining the Jost function, the occurrence of intruder orbitals is explained and it is possible to trace continuously the PS 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.

Conclusions: 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.

Figure 1

Potentials $\Sigma \left(r\right)$ and $\Delta \left(r\right)$ of ${}^{208}$Pb calculated using the relativistic mean-field model with the parameter set PC-PK1 [100]. These potentials can be approximated by Woods-Saxon (W.S.) potentials (dotted curves) or spherical square-well (S.W.) potentials (dashed lines) with a radius around 7 fm and depths around 650 and 66 MeV. The long tails in the proton case are due to the Coulomb interaction.

Figure 2

The Jost function ${\mathcal{J}}_{\kappa }\left(E\right)$ (in arbitrary unit) on the real $E$ axis for several pairs of pseudospin partners. The results for pseudospin $\tilde{s}=\pm 1/2$ are denoted as solid and dashed curves, respectively. The zero points representing bound states with $\tilde{s}=\pm 1/2$ are denoted as black and red dots, respectively.

Figure 3

Energies of bound states as well as resonances with $\tilde{l}=1,2,\cdots ,6$ in spherical square-well potentials with $C=-66$ MeV and $D=650$ MeV. The results with pseudospin $\tilde{s}=\pm 1/2$ are denoted as black and red lines, respectively. The bottom of the potential well $E=-66$ MeV and the threshold $E=0$ are shown as dotted lines. For each pseudo-orbital angular momentum $\tilde{l}$ the lowest level is an intruder state and has no pseudospin partner.

Figure 4

Widths of resonant states with $\tilde{l}=1,2,\cdots ,6$ in spherical square-well potentials with $C=-66$ MeV and $D=650$ MeV. The results with pseudospin $\tilde{s}=\pm 1/2$ are denoted as black and red lines, respectively. The bound-state threshold $E=0$ is shown as a dotted line.

Figure 5

Zeros of the Jost function ${\mathcal{J}}_{\kappa }\left(E\right)$ in spherical square-well potentials with different potential depth $C=-70,-60,\cdots ,0$ MeV for (a) $\tilde{l}=2$: $p_{3/2}$ ($\kappa =-2$) and $f_{5/2}$ ($\kappa =3$); and (b) $\tilde{l}=3$: $d_{5/2}$ ($\kappa =-3$) and $g_{7/2}$ ($\kappa =4$). When $C=0$, pseudospin doublets are degenerate and corresponding symbols overlap each other. The results with pseudospin $\tilde{s}=\pm 1/2$ are denoted as solid and open circles, respectively. When $C=-70$ MeV, energies of intruder states are below $-40$ MeV and not shown.

Figure 6

Energies of bound and resonant states for $p_{3/2}$ and $f_{5/2}$ with $\tilde{l}=2$ in spherical-well potentials as functions of the potential depth $C$. The results with pseudospin $\tilde{s}=\pm 1/2$ are denoted as solid and dashed curves, respectively. All the levels are paired off except for the lowest one. The bottom of the potential $E=C$ and the bound state threshold $E=0$ are shown as dotted lines.

Figure 7

The energy splitting between PS partners with the pseudo-orbital angular momentum $\tilde{l}=2$, $p_{3/2}$, and $f_{5/2}$, as a function of the potential depth $C$.

Figure 8

Widths of resonant states for $p_{3/2}$ and $f_{5/2}$ with $\tilde{l}=2$ in spherical-well potentials as functions of the potential depth $C$. The results with pseudospin $\tilde{s}=\pm 1/2$ are denoted as solid and dashed curves, respectively.

Figure 9

The width splitting between the PS partners with the pseudo-orbital angular momentum $\tilde{l}=2$ ($p_{3/2}$ and $f_{5/2}$) as a function of the potential depth $C$.