Antibound poles in cut-off Woods-Saxon and strictly finite-range potentials

The motion of $l=0$ antibound poles of the $S$ matrix with varying potential strength is calculated in a cut-off Woods-Saxon (WS) potential and in a potential that goes to zero smoothly at a finite distance and is exactly zero beyond [P. Salamon and T. Vertse, Phys. Rev. C 77, 037302 (2008)]. The pole positions of the antibound states as well as of the resonances depend on the cutoff radius, especially for higher node numbers. The starting points (at potential zero) of the pole trajectories correlate well with the range of the potential. The normalized antibound radial wave functions on the imaginary $k$ axis below and above the coalescence point have been found to be real and imaginary, respectively.

Figure 1

Imaginary part of the pole wave number as a function of the depth of the WS potential. For bound and unbound states $\mathrm{Im}\left(k\right)>0$ and $\mathrm{Im}\left(k\right)<0$, respectively.

Figure 2

Imaginary part of the pole wave number as a function of the depth of the SV potential.

Figure 3

Trajectories of the two $n=1$, $l=0$ poles in the WS potential with $V_0$ varied ($R_{\max }=15$ fm).

Figure 4

Normalized radial wave functions of antibound states in WS potentials. The $n=1$ ($j=2,3$) and $n=2$ ($j=4,5$) states were produced by $V_0=6.9692$ and 19.5 MeV, respectively. The wave numbers $k_j$ (in fm${}^{-1}$) are $k_2=-0.183\mathrm{i}$, $k_3=-0.188\mathrm{i}$, $k_4=-0.410\mathrm{i}$, and $k_5=-0.422\mathrm{i}$. The functions with $j=2,4$ are imaginary, while those with $j=3,5$ are real.

Figure 5

Trajectories of the odd-$n$, $l=0$ poles in the SV potential with $V_0$ varied. The even-$n$ trajectories are similar, except for $n=0$, which runs along the imaginary $k$ axis.

Figure 6

$\mathrm{Re}\left(k_n\right)$ values of starting points of resonance trajectories for the WS potential cut off at $R_{\max }=15$ fm fitted with a straight line

Figure 7

$n=7$ resonance trajectories for the WS potential cut off at different $R_{\max }$ values.