Green's function method for single-particle resonant states in relativistic mean field theory

Relativistic mean field theory is formulated with the Green's function method in coordinate space to investigate the single-particle bound states and resonant states on the same footing. Taking the density of states for free particles as a reference, the energies and widths of single-particle resonant states are extracted from the density of states without any ambiguity. As an example, the energies and widths for single-neutron resonant states in ${}^{120}\mathrm{Sn}$ are compared with those obtained by the scattering phase-shift method, the analytic continuation in the coupling constant approach, the real stabilization method, and the complex scaling method. Excellent agreements with these methods are found for the energies and widths of single-neutron resonant states.

Figure 1

Contour path to perform the integrals of the Green's function on the single-particle complex energy plane. The path is chosen to be a rectangle with width $\gamma$ and length ${\varepsilon }_{\mathrm{cut}}$ from ${\varepsilon }_{\mathrm{cut}}^{-}$ to ${\varepsilon }_{\mathrm{cut}}^{+}$. The red (gray) crosses denote the discrete single-particle states and the green (gray) thick line denotes the continuum. The dashed line is the Fermi surface $\lambda$.

Figure 2

Density of neutron states $n_{\kappa }\left(\varepsilon \right)$ for $s_{1/2}$ block in ${}^{120}\mathrm{Sn}$ obtained by the RMF-GF method with PK1 and space sizes $R_{\max }=15\mathrm{fm}$ (a), $20\mathrm{fm}$ (b), and $25\mathrm{fm}$ (c) respectively. Spectra with $\varepsilon >0$ are compared with $n_{\kappa }\left(\varepsilon \right)$ obtained with potentials $V=S=0$ (denoted by the short-dashed line). The dashed line is the continuum threshold. The inserts enlarge the density of neutron states in the energy range $\varepsilon \in [0,2]\mathrm{MeV}$.

Figure 3

The integrands for the density of states, $\mathrm{Im}[{\mathcal{G}}_{\kappa }^{\left(11\right)}(r,r;\varepsilon +i\epsilon )+{\mathcal{G}}_{\kappa }^{\left(22\right)}(r,r;\varepsilon +i\epsilon )]$, in Eq. (12) at energy $\varepsilon =-57.70$ (a), $0.30$ (b), and $2.00\mathrm{MeV}$ (c) for $s_{1/2}$ block in ${}^{120}\mathrm{Sn}$. Calculations are done with $R_{\max }=15$, 20, and $25\mathrm{fm}$. For comparison, integrands for $n_{\kappa }\left(\varepsilon \right)$ obtained with $V=S=0$ and $R_{\max }=25\mathrm{fm}$ are shown as the short-dashed lines.

Figure 4

The same as Fig. 2, but for $j_{15/2}$ block.

Figure 5

The same as Fig. 3, but for $j_{15/2}$ block.

Figure 6

Density of neutron states $n_{\kappa }\left(\varepsilon \right)$ for different blocks in ${}^{120}\mathrm{Sn}$ calculated by RMF-GF method with PK1 and $R_{\max }=20\mathrm{fm}$ (blue [gray] solid line) and compared with $n_{\kappa }\left(\varepsilon \right)$ obtained with $V=S=0$ (red [gray] dotted line). The dashed line represents the continuum threshold.

Figure 7

Energies $\varepsilon$ and widths $\Gamma$ of single-neutron resonant states $p_{1/2}$, $h_{9/2}$, $f_{5/2}$, $i_{13/2},i_{11/2}$, and $j_{15/2}$ in ${}^{120}\mathrm{Sn}$ obtained by the Green's function method in the framework of RMF theory with PK1 and NL3.