Bose-Einstein Condensation for a Class of Wave Functions

It is verified that the class of wave functions used in variational calculations for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ has Bose-Einstein condensation. These trial wave functions have the form of a product of pair functions. Two cases are considered: In the first the correlations in the wave function are assumed to be short range and in the second the correlations include also the long-range part due to the zero-point motion of phonons. The one- and two-dimensional system is briefly considered.