Critical analysis of the hard-sphere model for superfluidHe4. I.

The basic goal of this paper is to present the theory which stands at the basis of the calculations of the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ excitation spectrum reported by Wong and Huang. The exact $N$-particle hard-sphere pseudopotential is constructed through the method of non-local-field operators developed earlier. Keeping only the two-body interaction terms, this pseudopotential replaces the exact two-particle boundary conditions which is demonstrated here and thus has been proposed as a model potential for actual helium-helium interaction in the liquid state. A pair state of the Bogoliubov type is used to calculate the ground-state energy of such a system. An expansion-parameter scheme is presented which drastically simplifies the mathematics involved and justifies previous approximations employed by one of the authors. The expansion describes the dilute case by assuming a large portion of the particles in the zero-momentum state and the dense case by assuming only a negligible portion of the particles in the Bose-Einstein state.