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 E∼E⁰+(ℏ²∕2m)2πNϱa, where E⁰ is the energy of the noninteracting system and ϱ is the density. A similar formula holds in two dimensions (2D), with ϱa replaced by ϱ∕∣ln(ϱa²)∣. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.