Dilute Bose gases interacting via power-law potentials

Neutral atoms interact through van der Waals potentials, which asymptotically fall off as $r^{-6}$. The behaviors of dilute Boses gases can to a good approximation be described by the atom-atom scattering length $a_s$. However, as the system becomes more dense, corrections arise that depend on the characteristic length of the van der Waals potential. Making use of both essentially exact numerical calculations and semi-analytical solutions, we parameterize these corrections by analyzing the energetics of two- and few-atom systems under external harmonic confinement. We generalize these results to particles interacting through a longer-ranged potential, which asymptotically falls off as $r^{-4}$. Finally, we consider homogeneous systems interacting through different power-law potentials.