  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 $E \sim E^0 + (\hbar^2/2m) 2 \pi N \rho a$,
where $E^0$ is the energy of the non-interacting system and $\rho$ is the
density. A similar formula holds in 2D, with $\rho a$ replaced by $\rho
/|\ln(\rho a^2)|$. Obviously this 2D energy is not the expectation value of a
density-independent pseudopotential.
