# Green's function method for the spin and pseudospin symmetries in the single-particle resonant states

## TL;DR

This paper applies the Green's function method to study spin and pseudospin symmetries in single-particle resonant states within a Dirac equation framework, revealing the role of threshold effects and wave function similarities in nuclear physics.

## Contribution

It introduces a Green's function approach to analyze spin and pseudospin symmetries in resonant states, providing new insights into energy and width splittings in nuclear resonances.

## Key findings

- Threshold effects influence pseudospin doublet splittings.
- Reversed level structure observed in pseudospin doublets.
- Wave function similarities persist in resonant states.

## Abstract

We investigate the spin and pseudospin symmetry in the single-particle resonant states by solving the Dirac equation containing a Woods-Saxon potential with Green's function method. Taking double-magic nucleus $^{208}$Pb as an example, three spin doublets $3d$, $2h$, and $1j$ and three pseudospin doublets $3\tilde{p}$, $1\tilde{i}$, and $1\tilde{j}$ are obtained for the single-neutron resonant states. By analyzing the energy splittings, we find that the threshold effect plays an important role in resonant pseudospin doubles. Besides, there is a reversed level structure of pseudospin doublets in the continuum. Differently, all the width splittings of either the spin doublets or the pseudospin doublets are systematically positive and the splittings are very small except $1\tilde{j}$ doublet. Further studies show that the splittings of the energies and widths for the resonant (pseudo)spin doublets are independent. Besides, the similarity properties of the wave functions of the spin and pseudospin doublets still maintain well in resonant states.