Validity of the hartree-bogoliubov-theory in an exactly solvable model

Abstract

The structure and the validity of the Hartree-Bogoliubov (HB) theory are investigated within a special, exactly solvable model of the n-body problem, which was introduced previously and which contains only pairing and monopole forces. The HB theory is applied to this model in the variational form since there are convergence difficulties for the diagonalization procedure of solving the HB equation. For including the blocking effect the problems of even and odd numbers of particles are treated separately. For our model the HB equations yield solutions of essentially different type simultaneously (Hartree-Fock, BCS, mixed HB solutions). A comparison to the exact results shows that the properties of the energetically lowest solutions are satisfactory for ranges of model parameters where one of the two interaction types is overwhelming, whereas in the transition region there are large discrepancies (especially for the structure of the spectrum for the systems of an odd number of particles). The importance of the blocking effect is small and it can be taken into good account by introducing suitable constraints on the mean number of particles.