A study of pairing correlations for weakly-bound systems at the drip lines in a simple one-dimensional model

The physical case of Borromean nuclei is modelled in terms of an inert core plus two particles moving in a mean-field one-dimensional potential and interacting via a density-dependent residual delta interaction. The model assumes that, due to the vicinity of the drip line, all bound single-particle levels are occupied by the core particles and therefore all the unperturbed additional two-particle states lie in the continuum. The two-particle system becomes bound due to the action of a density-dependent residual delta interaction acting among the two particles. The problem is solved by introducing a discrete basis to account for the continuum part of the spectrum, imposing vanishing boundary conditions within a box. We calculate the particle-hole and particleparticle response functions to this system, relevant for describing break-up and two-particle transfer processes, and study their convergence as a function of the box radius. The peculiarities and the characteristics of the correlated weak-binding situation show up clearly in this one-dimensional case and are supposed to persist in the extension to the more realistic three-dimensional case.