Conclusions

Much is known about the analytic properties of one-particle wave functions in the low energy continuous spectrum when there is a real, virtual or quasistationary level with energy close to zero [68,69]. It was shown for a number of potentials, that the continuum wave functions in a wide range of ${\bf r}$-values have the ${\bf r}$-dependence which is remarkably close to that of the wave function for the zero-energy level [74,68,69]. Validity of this finding has been also discussed in Sec. 4.2 (see Fig. 3). As noticed by Migdal et al. [68], this approximate factorization of the continuum wave functions in the resonance region of one-particle phase space should simplify the computation of matrix elements. In the HF+BCS approach of Refs. [30,31], it is assumed that the analytical features of the S-matrix, which lead to the factorization property of the single-particle wave functions in the resonance region, remain valid in the presence of pairing correlations. Actually, this weak perturbation hypothesis for the S-matrix in the presence of pairing interaction remains not proved. To which extend the structure of single-particle resonances and the non-resonant continuum is affected by the presence of short-ranged correlations of the pairing type is the main problem which we have addressed in this work using the two-component quasiparticle wave-functions of the HFB approach and the PTG single-particle potential for which the S-matrix properties are known analytically. The detailed analysis performed in this work, shows that this weak perturbation assumption for the analytic structure of the S-matrix for the one-particle problem may be hazardous in many situations.

The comparison of the localization of HFB upper component of the quasiparticle states with the localization of the s_1/2 continuum PTG states exposed a complicate interplay between resonant and non-resonant HFB continuum which by no means can be approximated by the HF+BCS approximation spanned on the skeleton of the S-matrix resonances for the one-particle problem. The energy position of canonical states, which govern the pairing properties of the system, is not correlated with the presence and position of pole in the S-matrix for the one-particle problem. As a consequence, the pairing coupling of resonant and non-resonant s_1/2continuum is quite similar, and its magnitude depends on the single-particle localization in the interval of 2-3MeV above the Fermi surface.

When the single-particle poles are very close to the real axis in the momentum space ( $\Re(k)>0$), like in the case of very narrow high-j resonances deep inside the centrifugal barrier, the pairing interaction is too weak to perturb the single-particle pole structure and, hence, these resonances are not very different from the bound states.

In the intermediate case, like d_3/2-resonances studied in Sec. 5.2, the situation depends on the position of the single-particle resonance with respect to the top of the centrifugal barrier. In a typical case, however, norms of the lower HFB components closely follow the pattern of localizations in the corresponding single-particle continuum, i.e., localizations of the single-particle continuum states determine the strength of the pairing coupling of the HFB quasiparticle continuum. However, like for the s_1/2 continuum, none of the quasiparticle states can be used a as a single representative of the continuum phase space.

This research was supported in part by the Polish Committee for Scientific Research (KBN) under Contract No. 2 P03B 040 14, by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), DE-FG05-87ER40361 (Joint Institute for Heavy Ion Research), DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp. (Oak Ridge National Laboratory), by the NATO grant CRG970196, by the French-Polish integrated actions programme POLONIUM, and by the computational grant from the Interdisciplinary Centre for Mathematical and Computational Modeling (ICM) of the Warsaw University. The authors wish to thank W. Nazarewicz for the hospitality extended to them during the visit at ORNL where part of this work has been done and for the critical reading of the manuscript.