Wave functions

In the following we aim at analyzing the influence of weakly bound states and low-energy resonances on pairing properties of nuclei near neutron drip lines. Therefore, knowing the analytical solutions available for the PTG potential, we chose three sets of parameters such that the 3s_1/2 state is either weakly bound, or virtual, or there is a low-lying resonance in the s_1/2 channel. These three physical situations can be achieved by fixing parameters $\Lambda$ and s and shifting the depth parameter $\nu_{s{1\over2}}$. In the specific examples discussed below, we use the parameters listed in Table 2, where also the corresponding energies are given.

Wave functions of the resonant, virtual, and bound 3s_1/2 states are shown in Fig. 3. The resonant wave function is calculated at the energy equal to the real part of the pole shown in Table 2. Normalizations of the virtual and resonant wave functions are chosen so as to match the height of the first maximum of the bound wave function. The three wave functions illustrate properties of single-particle phase space in the situation where the 3s_1/2 state leaves the realm of bound states, dwells for a short time in a ghost-like zone of virtual states, and then reappears as a decent (although very broad) resonance. From the point of view of the asymptotic properties, pertinent to the scattering problems, these three wave functions are completely different. On the other hand, from the point of view of their structure inside the nucleus (see the inset in Fig. 3), they are almost exactly identical. In fact, in the scale of the inset, the resonant wave function cannot be distinguished from the bound one.

In Sec. 5 we also study several cases of different positions of the 2d_3/2 states. In particular, we use two values of the depth parameter $\nu_{d{3\over2}}$, which are listed in Table 2 together with the corresponding energies of the 2d_3/2 PTG resonance and virtual states. Whenever the spectrum in all partial waves is required, like in the HFB calculations below, we combine $\nu_{d{3\over2}}$=4.900 with all the three values of $\nu_{s{1\over2}}$, and $\nu_{s{1\over2}}$=5.034 with all values of $\nu_{d{3\over2}}$.