Sum rules and the momentum distribution in Bose-condensed systems

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