Ground-State Energy of a Many-Particle Boson System

The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\alpha =1-g\left(0\right)\mathrm{}\ll 1$ and the latter by $\beta =\frac{(N-N_0)}{N}\ll 1$, where $g\left(r\right)\mathrm{}$ is the radial distribution function and $N_0$ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\alpha$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\beta \ll 1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\delta$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.