The asymptotic properties of the two components of the HFB quasiparticle wave functions, and their dependence on and , were discussed in Refs. [17,5,18]. Here we complement this discussion by further elements, which pertain mainly to weakly bound systems where the Fermi energy is very small.
Assuming that = , which anyhow is always fulfilled in the asymptotic region, and neglecting for simplicity this trivial mass factor altogether, Eqs. (34) have for large the following form
In order to discuss this question, we note that the coupling terms can be considered as inhomogeneities of the linear equations (35), and therefore, asymptotic solutions have the form
We arrive here at the conclusion that the asymptotic form of may involve two terms and a more detailed analysis is needed before concluding which one dominates. To this end, we note that the coupling potential depends on the sum of products of lower and upper components, and therefore has a general form of
In order to illustrate the above discussion, we have performed the HFB calculations in Sn, where =0.345MeV and the lowest =0 quasiparticle state of =0.429MeV (for =30fm) leads to a very diffused coupling potential with small decay constant . Quasiparticle wave functions corresponding to the four lowest =0 quasiparticle states are shown in Fig. 1. One can see that the asymptotic forms of the second components of the two lowest quasiparticle states are not affected by the second term , at least up to 30fm. Only the third and fourth states switch at large distances to the oscillating asymptotic forms with a smaller decay constant. This happens at rather large distances where the densities are anyhow very small. Hence the change in the asymptotic properties does not affect any important nuclear observables.
In all cases that we have studied, the practical importance of the second term is negligible. However, its presence precludes simple analytic continuation of the wave functions in the asymptotic region. We would like to stress that the asymptotic forms discussed in this section are numerically stable, and that they are unrelated to the numerical instabilities discussed in Sec. 5.3 below.