1. A sufficient condition for the validity of the following considerations is that the potential be asymptotically integrable, namely vanish asymptotically faster thanr −4, whered is the dimensionality of the space (d=3 for ordinary space). This we assume hereafter.
2. When compressed to high density the system might crystallize, namely a configuration characterized by periodic density fluctuations might be energetically favoured over the homogeneous and isotropic one. This would happen if the potential were strongly repulsive at short range (see, for instance,E. H. Lieb andM. de Llano:Phys. Lett.,37 B, 47 (1971)). The model that we shall consider is not adequate to reproduce such a behaviour. Another phenomenon that may occur when the system is compressed is phase separation; as we will see, the model of this paper can display this phenomenon.
3. Here, and always in the following, we assume the particles to obey Bose (or Boltzmann) statistics. For fermions, the Pauli principle prevents to some extent the particles from congregating at one point, but, at least in ordinary (3-dimensional) space, not to a sufficient extent to modify the qualitative features of the phenomenon of collapse. See, for instance,D. Ruelle:Statistical Mechanics (Rigorous Results), Chap. 3 (New York, N. Y., 1969);F. Calogero andYu. A. Simonov:Rigorous contraints that nuclear forces must satisfy to be consistent with the saturation property of nuclear binding energies, inThe Nuclear Many-Body Problem, edited byF. Calogero andC. Ciofi Degli Atti,Proceedings of the Symposium on Present Status and Novel Developments in the Nuclear Many-Body Problem, Rome, September 1972 (Bologna, 1974).
4. As implied by this analysis, a necessary condition to prevent collapse is positivity ofV(0). Because collapsed configurations other than that in which all particles congregate at the same point are also possible, this condition is not sufficient to prevent collapse; indeed another condition that is also necessary to prevent collapse is positivity of the volume integral ofV(r) (3). A condition on the interparticle potential that is sufficient to prevent collapse is positivity of its Fourier transform (note that this does not require positivity of the potential itself for all values ofr); see, for instance,F. Calogero, Yu. A. Simonov andE. L. Surkov:Phys. Rev. C,5, 1943 (1972).
5. We are assuming no crystallization or phase separation to occur, so that the condensate is uniform and its bulk density is well defined; otherwise one should talk of its average density.