Eigenphase shift decomposition of the random-phase approximation strength function based on the Jost-RPA method

The S-matrix which satisfies the unitarity, giving the poles as random-phase approximation (RPA) excited states, is derived using the extended Jost function within the framework of the RPA theory. An analysis on the correspondence between the component decomposition of the RPA strength function by the eigenphase shift obtained by diagonalization of the S-matrix and the S- and K-matrix poles was performed in the calculation of the ${}^{16}\mathrm{O}$ quadrupole excitations. The results show the possibility that the states defined by the eigenphase shift can be expressed as RPA-excited eigenstates corresponding to the S-matrix poles in the continuum region.

Figure 1

The isoscalar and isovector RPA strength functions for the quadrupole excitation of ${}^{16}\mathrm{O}$ were shown by the solid blue and red curves in the top and middle panels, respectively. The dashed line shows the RPA strength function calculated with ${\kappa }_{np}=0$, multiplied by factors 0.05 and 0.4 in the top and middle panels, respectively. The trajectories of poles [(a)–(f)] obtained by solving Eq. (21) varying the constant parameters $f_1$ and $f_2$ multiplied to the residual interaction are shown in the bottom panel. The open circle ($\circ$), cross ($\times$), and filled circle ($•$) symbols show the position of poles which are calculated by the RPA (i.e., $f_1=1$, $f_2=1$), RPA with ${\kappa }_{np}=0$ ($f_1=1$, $f_2=0$) and unperturbed($f_1=0$, $f_2=0$) solutions, respectively (see Table 3).

Figure 2

The same figure with Fig. 1 but calculated without Coulomb interaction. The trajectories of poles are calculated by solving Eq. (101), the positive and negative signs for the isoscalar and isovector modes, respectively (see Table 4).

Figure 3

Eigenphase shift ${\delta }_{\alpha }^{\left(1\right)}$ obtained by diagonalizing the ${\mathcal{S}}^{\left(1\right)}$ matrix using Eq. (75) (bottom panel) and decomposition of the RPA strength function by the eigenphase shift components (second and third panels). [(a)–(f)] in the upper figure are the S-matrix poles shown in Fig. 1 and Table 3. The imaginary part of the poles is represented by error bars. [(i)–(v)] show the positions of the K-matrix poles appearing on the real axis of energy. The energy of the K-matrix pole is equivalent to the energy when the eigenphase shift ${\delta }_{\alpha }^{\left(1\right)}$ crosses $\pi /2$ (see Table 5).

Figure 4

The same figure as Fig. 3 but calculated without Coulomb interaction.

Figure 5

Magnified view of the energy region from 16 to 19 MeV in Fig. 3.

Figure 6

Magnified view of the energy region from 16 to 19 MeV in Fig. 4.

Figure 7

The isovector transition density ${\rho }_{\delta ,\tau =1}^{\left(1\right)}$ and isoscalar transition density ${\rho }_{\delta ,\tau =0}^{\left(1\right)}$ at the first K-matrix pole energy of the eigenphase shift giving the K-matrix poles [(i) and (${\mathrm{i}}^{\prime }$)].

Figure 8

The isovector transition density ${\rho }_{\delta ,\tau =1}^{\left(1\right)}$ and isoscalar transition density ${\rho }_{\delta ,\tau =0}^{\left(1\right)}$ at the K-matrix pole energy of the eigenphase shift giving the K-matrix poles [(ii–v) and (${\mathrm{ii}}^{\prime }–{\mathrm{vi}}^{\prime }$)].

Figure 9

The ${\delta }_2^{\left(1\right)}$ and ${\delta }_5^{\left(1\right)}$ components of the transition densities (IS and IV modes) at K-matrix poles (ii), (iii), (${\mathrm{ii}}^{\prime }$), and (${\mathrm{iii}}^{\prime }$), respectively.