Exact results on the two-particle Green’s function of a Bose-Einstein condensate

Starting from the Dyson-Beliaev and generalized Gross-Pitaevskii equations with an extra nonlocal potential, we derive an exact expression of the two-particle Green’s function $\underset{̱}{\mathcal{K}}$ for an interacting Bose-Einstein condensate in terms of unambiguously defined self-energies and vertices. The formula can be a convenient basis for approximate calculations of $\underset{̱}{\mathcal{K}}$. It also tells us that poles of $\underset{̱}{\mathcal{K}}$ are not shared with (i.e., shifted from) those of the single-particle Green’s function, contrary to the conclusion of previous studies.

Figure 1

Irreducible vertices of Eq. (15).

Figure 2

Diagrammatic representation of Eq. (21). Every long straight line in the second equation denotes $\widehat{G}$.