Ground-state energy of the low-density Fermi gas

Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of N fermions interacting with a short-range, positive potential of scattering length a. For spin-1∕2 fermions, this is EE^0+ (^2∕2m) 2, where E^0 is the energy of the noninteracting system and is the density. A similar formula holds in two dimensions (2D), with replaced by ∕ (a^2). Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.