Perspectives and outlook

The states at positive energies having maximum localization inside the region of potential well, preserve a remnant of the eigenvalue structure [68,69]. Whether virtual or resonant, these low lying states are likely to play an important role in the low-energy phenomena. For example, the neutron direct radiative capture process, which depends on properties of nuclear potential both at the surface and in the interior regions, is a unique tool to investigate properties of nuclear wave functions. Similar information may also be derived from the inverse process, i.e., the Coulomb dissociation process, and both those processes are going to be very important in extracting the information about nuclei close to the drip line. Raman et al. [70] have shown how strongly the details of a neutron-nucleus potential affect the capture mechanism for s-neutrons. (Extension of this analysis to higher partial waves has been done recently by Mengoni et al. [71].) This is also important in many astrophysical issues, e.g., the inhomogeneous Big Bang model or the neutron poison problem.

Low-lying resonant states may influence the pairing correlations enhancing the binding energy of halo nucleons (neutrons). In this work we have seen that a presence near threshold of the L=0 poles of the S-matrix corresponding to bound, virtual, and resonance states, may influence the pairing correlations and increase the localization of scattering wave functions in the narrow range of $\Delta \epsilon\simeq$2-4MeV above the threshold. Hence, the pairing interaction induces a subtle rearrangement effect in the structure of scattering states which modifies the strength of pairing field and the binding energy of affected nucleons close to the Fermi surface what, in turn induces new rearrangement in the structure of low lying scattering states, etc. This effect can be described only in the self-consistent treatment of pairing correlations.

For the HF potentials where the resonance width $\Gamma$ tends to a finite value when the resonance energy $\epsilon$ tends to zero, in the physical situation corresponding to the drip lines, i.e., when the nucleon (neutron) separation energy tends to zero ( $S_n \rightarrow 0$), and hence $-\lambda_n$- $\Delta\simeq$0, the standard quasiparticle picture [72] may be questionable. For the quasiparticle picture to be valid, the appreciable fraction of the single-particle strength should remain in the quasiparticle excitations and, moreover, the lifetime of excitations about the Fermi surface should be relatively long. Whether the single-particle strength is concentrated in the quasiparticle excitations depends on the strength of the residual interaction mixing the single-particle strength into more complicated configurations. But even when the residual interaction is strong, one can always assure in normal systems that the quasiparticle lifetime is long, i.e., the spreading of the quasiparticle strength function is small. In those systems, width of a quasiparticle state $\Gamma(k)\sim(k-k_F)^2$ can be made small by letting k approach k_F [73]. This result is independent of details of the interaction and follows from the Pauli principle and the phase-space considerations. However, as demonstrated in the example of the PTG potential, the states above the Fermi surface of k_F$\simeq$0 may have a finite width in the limit of $k\rightarrow k_F$. In spite of the fact that the PTG centrifugal barrier is different from the physical centrifugal barrier, this feature should be seen in the structure of weakly bound nuclei near the drip line, at least for systems where L=0 states are present near threshold.

This particular aspect does not seem to be modified by including the pairing correlations within the HFB method. Consequently, the perturbation theory may not be applicable in these systems, and various kinds of instabilities caused by the residual correlations are expected. These instabilities may change the initial HF(B) vacuum in these nuclei nonperturbatively. In other words, one may expect that the spectrum of excitations in the unperturbed HF(B) system may not be compatible with the spectrum of excitations in the HF(B) system perturbed by the residual correlations. This brings about a new and challenging aspect into the experimental studies of shell structure and excitations close to the drip line: The atomic nuclei at the drip-line may provide a new kind of quantum open system with yet unknown properties of its excitation spectrum, resulting from the strong coupling between bound interior states and the environment of scattering states. If discovered, its microscopic description may call for new techniques in solving the quantum many-body problem. Similar coupling between the localized quasi-stationary states and the scattering states has already been recognized to be responsible for the unusual features of the surface of nuclear potential [58].

One of the most interesting aspects of correlations in drip-line systems is the question of multipole instabilities and deformations. A basis to study such phenomena should be provided by including the HFB pairing correlations in deformed states. This is a very difficult task, and only recently first attempts of such solutions become available [17,18,34]. The problem here is related to solving the HFB equations in a deformed coordinate-space representation, in a situation where variational methods are not applicable because the HFB equation has a spectrum unbounded from below. However, in our opinion, only such an approach may provide a sound basis of quasiparticle states in which other correlations (perturbative or not) may further be investigated.

It is essential to distinguish between five different aspects of the pairing coupling to the continuum phase space. First, enough continuum has to be included to cover the zone around the Fermi level, which is reasonably larger than the pairing gap. For gaps of about 1MeV the zone which is 5MeV wide is often used, cf. Ref. [17]. Second, the low-lying continuum with large localizations has to be taken into account. In cases studied here (Fig. 5), the zone of 5MeV seems to be enough, however, this aspect strongly depends on the shape and depth of the single-particle potential, and the safe limit should probably be at least twice larger. Third, enough continuum should be included so as the contributions N_n to particle number become small, and quasiparticle states entering and leaving this zone do not cause significant changes. A limit of N_n$\simeq$0.001-0.0001 is probably the least one can get away with, which already requires going up to 10-15MeV into the continuum, see Fig. 8. Fourth, all quasiparticle states which significantly contribute to the canonical states (i.e., those which have large spectral amplitudes [12]) should be included, which requires 10-20MeV of the continuum [12]. Finally, a coupling between particle-type and hole-type quasiparticle states has to be taken into account. Although this coupling is not responsible for the physical widths of deep hole states, cf. discussion in Ref. [12], it is present in the HFB equations, and affects continuum solutions. This aspect requires taking continuum up to energies exceeding the depth of the single-particle potential, i.e., to about 40-50MeV, and such a prescription [6] has been used here. Needless to say that the above discussion concerns the HFB continuum; analyses based on the BCS method, which use the single-particle continuum, have other properties, and often diverge with increasing continuum cut-off energy.