The superfluid DFT based on self-consistent HFB has already become the standard tool to describe pairing correlations in atomic nuclei. Such framework has been implemented in numerous approaches aiming at a consistent description of particle-hole and particle-particle channels, and it is gradually replacing a much simpler original BCS theory. This is so, because in finite systems like nuclei, spatial dependence of particle and pairing fields has to be properly described, especially in the nuclear periphery of weakly bound isotopes. In this respect, the BCS theory and its different flavors are manifestly deficient[18,19,20,21].
In this study, we aimed at presenting some basics of the local superfluid DFT along with several aspects of it related to advanced current applications. There are, of course, numerous aspects of the HFB theory that we could not cover in this limited overview. First, there have been many applications of the HFB theory using finite-range interactions, which imply nonlocal pairing fields. While they are significantly more difficult to treat, they do not lead to ultraviolet divergencies. Based on the current description of the limited set of nuclear observables related to pairing, it is difficult to judge whether the finite range is essential. In fact, one can understand finite-range interactions in terms of regularized local functionals. Second, we did not discuss various issues related to the restoration of particle-number symmetry. Effects of particle-number nonconservation are probably little significant in heavy nuclei, but they may become crucial for some observables and in specific systems, like, for example, nuclei with only few particles in valence shells. Third, we could not cover subjects related to the treatment of pairing in high-spin states where the broken time-reversal symmetry precludes the use of the BCS theory. Fourth, pairing correlations impact nuclear dynamics in a profound way. Recently, there have been many exciting developments related to the treatment of small- and large-amplitude collective motion in weakly-bound superfluid nuclei. Finally, we did not discuss details of the HFB theory applied to the isoscalar pairing. This channel becomes essential in nuclei with almost equal numbers of protons and neutrons, and numerous applications of the HFB theory to this case exist in the literature, see Refs.[25,26] Many of these topics are discussed in other contributions contained in this Volume.[3]