Pseudospin symmetry in single-particle resonances in spherical square wells

Abstract

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.

Hidden pseudospin and spin symmetries and their origins in atomic nuclei

“…This has been done in Ref. [249] in which were found a threshold effect in the energy splitting and an anomaly in the width splitting of pseudospin partners when C varies from zero to a finite value. As the depth of the singleparticle potential becomes larger, the PSS is broken more and a threshold effect in the energy splitting appears which can be seen in the left panel of Fig.…”

“…There are still many open problems concerning the pseudospin and spin symmetries in single-particle resonant states, as listed in Refs. [248,249].…”

“…Recently, Lu, Zhao, and Zhou [247] gave a rigorous verification of the pseudospin symmetry in single-particle resonant states. They have shown that the pseudospin symmetry in single-particle resonant states in nuclei is exactly conserved under the same condition for the pseudospin symmetry in bound states, i.e., Σ(r) = 0 or dΣ (r)/dr = 0 [247][248][249]. The exact conservation and breaking mechanism of the pseudospin symmetry in single-particle resonances for spherical square-well potentials have been investigated, in which the pseudospin symmetrybreaking part can be separated from other parts in the Jost functions.…”

“…To understand more deeply the pseudospin symmetry in single-particle resonant states, in Ref. [249], the exact conservation and breaking mechanism of the pseudospin symmetry in single-particle resonant states in square well potentials were extensively studied. A threshold effect in the energy splitting and an anomaly in the width splitting of pseudospin partners were found when the depth of the potential varies from zero to a finite value.…”

Pseudospin symmetry in nuclear structure and its supersymmetric representation

“…They discovered that the pseudospin symmetry in the single-particle resonant states is exactly conserved under the same condition discussed for the bound states, i.e., Σ(r) = 0 or dΣ(r)/dr = 0. By examining the zeros of Jost functions corresponding to the lower component of Dirac spinor, general properties of pseudospin-symmetry breaking in energy and width were examined, and the pseudospin-symmetry-breaking part can be separated from other parts in the Jost functions [122].…”