Ground-State Energy of Bose-Einstein Gas with Repulsive Interaction

The explicit evaluation of the ground-state energy of a Bose gas interacting through a two-body repulsive potential as an expansion essentially in powers of the density of the particles and the scattering length is given. It is shown that the problem can be treated by using diagrammatical analysis up to any orders. The energy was investigated up to the fourth order in the ordinary sense of a perturbation expansion in terms of the scattering length, but including terms up to infinite order for special classes of diagrams. The energy obtained contains in general a $\Sigma n^{}{C}_n{\rho }^n\ln \rho$ ($n\geqq 2$, half-integer and integer) dependence, and the coefficient $C_2$ is evaluated. A self-consistent treatment is given following this analysis and is shown to lead to the formulation of Beliaev.