Introduction

In the theoretical description of drip-line nuclei, the residual coupling between bound states and the continuum is an essential element of the physical situation. Standard description of many-fermion systems [1] most often invokes the concept of the Fermi sphere of occupied states, and correlations of particles occurring mostly in a narrow zone of the phase space around the Fermi surface. It is obvious that whenever the Fermi energy is close to zero, which by definition is the case in weakly-bound systems, the zone of correlated states must incorporate the phase space of particle continuum.

Among different types of correlations which are important in nuclei, in drip-line nuclei pairing plays a singular role, because the intensity of pairing correlations is a determining factor in establishing position of the last bound nucleus. Indeed, in any isotopic chain, the lightest unbound nucleus corresponds to an odd-N system in which the pairing energy $\Delta$, missing in the binding of the odd neutron, equals to the loss of the mean-field binding when this neutron is shaken off, see e.g. discussion in Refs. [2,3,4]. Therefore, study of pairing correlations in weakly bound systems is since many years one of the leading subjects of analyses predicting properties of very neutron rich nuclei.

After early analyses, see Refs. [5,6,7,8,9,10,11] and references cited and reviewed in Refs. [12,13], the field has now become very active, due to projected developments in radioactive ion beam facilities throughout the world. In particular, methods based on using the Skyrme effective interaction and on the Hartree-Fock-Bogoliubov (HFB) approximation have been applied to both spherical [12,13,14,15,16] and deformed [17,18,19] drip-line nuclei. Spherical drip-line nuclei have also been studied by using the Gogny interaction and the HFB method [12,16]. In addition, methods based on the relativistic mean field approach [20], with pairing correlations included within the Hartree-Bogoliubov approximation, have been extensively used to study various phenomena in spherical drip-line nuclei [21,22,23,24,25,26,27,28]. Structure of pairs have also been studied by solving the two-body problem exactly, including the continuum effects [29], and the influence of particle resonances on pairing properties have been analyzed in terms of the BCS method [30,31].

In the framework of the mean-field approximation, pairing correlations are consistently included within the variational Hartree-Fock-Bogoliubov (HFB) approach [32]. Such an approach is appropriate for studying bound nuclei, i.e., systems with negative Fermi energy, in which the effects of continuum amount to ensuring that the obtained many-body states are stable with respect to virtual excitations of particles and pairs of particles to positive-energy phase space. In this study we do not consider scattering problems, for which the wave functions are not localized and describe physical situations of particles being scattered off, or emitted by nuclei. On the other hand, bound states, however weakly bound, are always localized and discrete, and such are the basic features of the approach discussed in the present study. A properly executed variational theory, and such is the HFB approach, always yields localized bound states. Moreover, when solved in the coordinate representation, the HFB method takes fully into account all the mean-field effects of coupling to the continuum [6,9,12].

The HFB equations are relatively easy to solve in the matrix representation, i.e., after expanding the quasiparticle wave functions on a suitable basis. When the harmonic oscillator (HO) basis is used, the method is inappropriate for a description of weakly-bound states, because of an incorrect asymptotic properties of the HO wave functions [12]. However, recently developed methods, which use the so-called transformed harmonic oscillator basis [27,33], combine the simplicity of the basis expansion with the corrected asymptotic behavior. In this way, large-scale deformed HFB calculations in drip-line nuclei become recently possible [34]. Another method proposed [35] for a solution of the HFB equations relies on using the natural orbitals; although very promising in principle, it has not been fully implemented in practice yet.

Apart from these attempts, most of the HFB calculations in drip-line systems performed to date use the coordinate representation. Most of these studies have been restricted to spherical symmetry, for which the coordinate-space problem amounts to solving one-dimensional equations. Three-dimensional solutions for deformed nuclei have also been obtained by using hybrid methods of solving the HFB equations in the basis of coordinate-space Hartree-Fock (HF) wave functions [17,18,19]. All such algorithms use wave functions approximated on spatial grids of points, and the continuum discretized by suitable large-box boundary conditions. Numerical stability and convergence properties of these methods have been thoroughly tested, and are considered to be sufficient for the current mean-field applications, however, more efficient techniques, using the so-called basis-spline Galerkin lattices [36] are also being constructed.

In view of the fact that pairing properties of nuclei near the stability valley are most often treated within the BCS approximation, there have been numerous attempts of using this simpler method in drip-line nuclei too, see e.g. Refs. [37,38,39,30,31]. Various tests show [6,12,40,41,42], however, that such an approach is often unstable and/or divergent. A meaningful use of the BCS approximation to the HFB method requires especially careful and conscious treatment, which may, in fact, cost more effort than a straightforward application of the more involved, but certain HFB method itself.

Resonance contribution into the HF+BCS equations was recently studied in Refs. [30,31]. This approximate method can take into account the effect for well separated resonances, when the resonance energies do not depend strongly on the box size or the cutoff procedure. More refined treatments of particle continuum have also recently become available [43,44], although they have not yet been combined with pairing correlations. On the other hand, intuition gained in solving the BCS problem in nuclei tells us that low-energy continuum states must more generally contribute to pairing correlations of drip-line nuclei. In the present paper we aim at quantifying this influence in the framework of the HFB method.

The full-scale HFB calculations and/or the analytically soluble models with realistic potential shapes are invaluable tools for understanding phenomena associated with the continuum in weakly bound systems. In this work, we shall employ the Pöschl-Teller-Ginocchio (PTG) average potential, which is an extension of the textbook Pöschl-Teller potential [45,46,47], proposed by Ginocchio [48]. It belongs to a broader class of analytically solvable potentials proposed some time ago by Natanzon [49]. The PTG potential has the main features of the nuclear mean field, namely, flat bottom, diffused surface, and asymptotic freedom (i.e., it exponentially vanishes at large distances). For this potential, analytical solutions are available for wave functions and energies of the single-particle bound states and resonances. These advantages make it very useful for applications to nuclear structure problems [50,51].

In Sec. 2, we recall the most important features of PTG potential, in particular those concerning the parametrization of the diffuse surface. Sec. 3 presents some essentials of resonance phenomena and discusses their relation with poles of the S-matrix. In Sec. 4, details of the single-particle wave functions and single-particle resonances in the PTG potential are recalled and, in particular, properties of wave function corresponding to weakly bound, virtual and resonance states are discussed. Sec. 5 contains main results of this work. Here results of the HFB calculations with the PTG input are discussed, and the pairing coupling to single-particle states in the continuum is analyzed. Finally, Secs. 6 and 7 summarize perspectives of further investigations and the main conclusions we draw from this study, respectively.