Sum rules and the momentum distribution in Bose-condensed systems

The momentum distribution ${\tilde{n}}_p$ of atoms in a Bose-condensed system is computed in the long-wavelength limit $p\to 0$. 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 ($\mathrm{cp}\ll k_{B}T$) the correction to ${\tilde{n}}_p=\frac{n_0{m}^2}{{\rho }_s{p}^2}$ involves the off-diagonal self-energy ${\Sigma }_{+-}(\overrightarrow{\mathrm{p}},\omega =0)$. In calculating ${\tilde{n}}_p$ in the limit $\mathrm{cp}\gg k_{B}T$, we include first-sound as well as second-sound contributions.