The Skyrme Hartree-Fock-Bogolyubov equations

The HFB approximation is based on the use of a trial variational wave function which is assumed to be an independent quasiparticle state . This state, which mixes different eigenstates of the particle number operator, is a linear combination of independent particle states representing various possibilities of occupying pairs of single particle states. Following the notations and phase convention of [5] we define the particle and pairing density and matrices by

(1) |

(2) |

where the operators and create and annihilate a nucleon at the point having spin and isospin . The symmetry properties of and as well as the relation between and the pairing tensor (defined for example in [10]) are discussed in [5].

The variation of the energy expectation value with respect to and under the constraints and (for neutrons and protons) leads to the Hartree-Fock-Bogolyubov equation which reads in coordinate representation

where the particle and pairing fields are given by

(3) |

Once again we refer the reader to the article [5] for the discussion concerning the quasiparticle spectrum, its symmetries and the relations between the components of the HFB spinors and the densities.

- Local densities
- Spherical symmetry
- The Hartree-Fock-Bogolyubov energy
- The Hartree-Fock-Bogolyubov mean fields
- The Hartree-Fock-Bogolyubov equations
- Asymptotic properties of the HFB wave functions
- Pairing correlations and divergence of the energy