Particle densities for dilute hard-sphere Bose or Fermi gases in an external potential

We prove that if the diameter of a hard-sphere is much smaller than the size of an external potential, the s-wave pseudopotential reduces to the Huang-Yang s-wave pseudopotential. We obtain the first-order virial expansions of particle densities for dilute hard-sphere Bose or Fermi gases in an arbitrary external potential. In the absence of an external potential, the results reduce to the Huang-Yang-Luttinger and Lee-Yang virial expansions. In the quasi-classical limit, the results reduce to the results of the local density approximation.