Commentary on self-consistent cranking plus RPA

A recent paper of Almehed et al (2003 J. Phys. G: Nucl. Part. Phys. 29 2193) utilizing the self-consistent cranking plus RPA method revealed a misunderstanding that should have been settled more than two decades ago. As a step towards rectification, the present paper shows, first of all, that the approach of Reinhardt and that of Marshalek can agree, providing that the RPA corrections are consistently treated as small compared to the mean field. Next, the two methods are tested on a couple of analytically solvable models. The first is a transparent two-dimensional many-boson model that is a generalization of one mentioned by Almehed et al, which is treated in detail going one order beyond the RPA. It is shown that the approach of Marshalek then yields exact results. As an outgrowth of the treatment, a novel representation of bosons rotating in two dimensions is briefly discussed. The second model is a diatomic-molecule potential model, which serves to illustrate some issues pertaining to three dimensions and also the effect of backbending