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 $E\sim E^0+({\hslash }^2∕2m)2\pi N\varrho a$, where $E^0$ is the energy of the noninteracting system and $\varrho$ is the density. A similar formula holds in two dimensions (2D), with $\varrho a$ replaced by $\varrho ∕\mid \ln \left(\varrho a^2\right)\mid$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.