Sum rules and the momentum distribution in Bose-condensed systems

Abstract

The momentum distribution ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}$ of atoms in a Bose-condensed system is computed in the long-wavelength limit $p\ensuremath{\rightarrow}0$. We use frequency-moment sum rules for the single-particle Green's function, including a generalized version of Wagner's sum rule appropriate to hard-core potentials. We show that at finite temperatures ($\mathrm{cp}\ensuremath{\ll}{k}_{B}T$) the correction to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}=\frac{{n}_{0}{m}^{2}}{{\ensuremath{\rho}}_{s}{p}^{2}}$ involves the off-diagonal self-energy ${\ensuremath{\Sigma}}_{+\ensuremath{-}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},\ensuremath{\omega}=0)$. In calculating ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}$ in the limit $\mathrm{cp}\ensuremath{\gg}{k}_{B}T$, we include first-sound as well as second-sound contributions.