Virtual-state character of the two-body system in the complex scaling method

In association with the property of the $1/2^{+}$ state observed just above the ${}^8\mathrm{Be}\left(0^{+}\right)+n$ threshold energy in ${}^9\mathrm{Be}$, we investigate the photodisintegration cross section of an $s$ state in a simple schematic two-body model using the complex scaling method. The photodisintegration cross section, continuum level density, scattering phase shifts, and scattering length are discussed in relation with the virtual state. These scattering observables show strong divergences when a virtual state is located near the physical scattering region.

Figure 1

Trajectories of the $s$-wave poles in momentum ($k$) and in energy ($E$) planes.

Figure 2

The energy level diagram of the two-body potential model simulating ${}^9\mathrm{Be}$. The dotted line represents the threshold energy.

Figure 3

The pole trajectories for the $J^{\pi }=0_1^{+}$, $1^{-}$, and $0_2^{+}$ states. The circles, triangles, and squares display the eigenvalues of the $J^{\pi }=0_1^{+}$, $1^{-}$, and $0_2^{+}$ states, respectively, calculated as changing the potential strength $V_1$.

Figure 4

Photodisintegration cross sections due to the $E1$ transition calculated with the two-body potential model with the strength $V_1=-1.42$ MeV. The arrow indicates the threshold energy which is shown in the Fig. 2 as a dotted line.

Figure 5

Calculated continuum level density ${\mathrm{\Delta }}_{\theta }^N\left(E\right)$ at the strengths of $V_1=-1.42$ MeV and $-1.43$ MeV. The black solid and red dashed curves are the results at $V_1=-1.42$ MeV and $-1.43$ MeV, respectively.

Figure 6

Calculated phase shifts of the $0^{+}$ state for $V_1=-1.42$ MeV and $-1.43$ MeV. The solid and open solid curves are phase shifts calculated at $V_1=-1.42$ MeV and $-1.43$ MeV, respectively. The scale of the graph was magnified in the inset.

Figure 7

Scattering length of the $0^{+}$ state calculated for $V_1=-1.2$ MeV and $-1.5$ MeV in the CSM. The horizontal dotted line indicates $a_s=0$ and the vertical broken line shows a border where the $a_s$ changes sign.

Figure 8

The difference between the continuum level densities for $V_1=-1.42$ MeV and $-1.43$ MeV. Ideally, the continuum level density calculated by the virtual state ${\mathrm{\Delta }}_{\theta }^{\mathrm{virt}}\left(E\right)$.

Figure 9

The phase shift of the virtual state, calculated from the virtual-state continuum level density ${\mathrm{\Delta }}^{\mathrm{virt}}\left(E\right)$.