Results

To test the applicability of our method near the neutron drip line, we performed calculations for $^{174}$Sn, which the Skyrme parameter set SkM$^\ast$ with volume-type delta pairing interaction places very close to the two-neutron drip line. The results in the isoscalar $0^+$ channel are shown in Fig. 1, which displays three curves calculated within different single-particle spaces. We include single-particle states for which occupation probabilities are larger than a cutoff parameter $v_{\rm crit}^2$ (which is set to a very small value so that we omit little of physical significance) if the system is paired in the HFB ground state, or those for which the Hartree-Fock (HF) energies are lower than a cutoff parameter $\varepsilon _{\rm crit}$, if the system is unpaired. In $^{174}$Sn, the neutrons are paired, and protons are unpaired in the HFB calculation. Figure 1 demonstrates that our solution converges when we make $v_{\rm crit}^2$ small enough and $\varepsilon _{\rm crit}$ large enough. Since the neutrons of $^{174}$Sn are paired, there is a spurious state associated with particle-number nonconservation. We checked that the transition strengths for the particle-number operator are smaller than 10$^{-5}$ to the real excited states.

The isoscalar $1^-$ mode is challenging technically because of spurious center-of-mass motion; a careful calculation is necessary to accurately separate the spurious state from real excited states. In calculations that are not fully self-consistent, the strength is often corrected by including a term $-\eta rY_{1M}(\Omega)$ (where $\eta = (5/3)\langle r^2\rangle$ with $\langle r^2\rangle$ the mean value in the HFB ground state) in the isoscalar-dipole transition operator [4]. We performed calculations of the strength functions with and without the correction term and obtained identical results for real excited states; our $1^-$ solutions are therefore essentially free from contamination. In a perfect calculation, the spurious state would have zero energy and the correction term would remove strength only from this state. In our calculation of $^{174}$Sn ($^{120}$Sn), even though the spurious state energy is 0.319 (0.713) MeV, the correction removes almost no strength except from this spurious state. This check is important for proving that the strong enhancement of strength at low energy in nuclei near the neutron drip line, illustrated in Fig. 2, is not an artifact of the calculation.

Finally we mention that the energy-weighted sum rules of the $0^+$, $1^-$, and $2^+$ modes of $^{120,174}$Sn are satisfied with errors of $\pm1$ % at most.

Figure 1: Strength functions in the isoscalar $0^+$ channel of $^{174}$Sn. The thick (thin) line was obtained with $v_{\rm crit}^2$ = 10$^{-16}$ (10$^{-8}$), for the neutrons, and $\varepsilon_{\rm crit}$ = 200 (100) MeV, for the protons. The result with $v_{\rm crit}^2$ = 10$^{-12}$ and $\varepsilon_{\rm crit}$ = 150 MeV is almost identical to the thick line.

Figure 2: Strength function in the isoscalar $1^-$ channel of $^{174}$Sn (upper) and $^{120}$Sn (lower). The low-energy strength is greatly enhanced in the drip-line nucleus.