Density functional theory of the trapped Fermi gas in the unitary regime

We investigate a density‐functional theory (DFT) approach for an unpolarized trapped dilute Fermi gas in the unitary limit. A reformulation of the recent work of T. Papenbrock [1] in the language of fractional exclusion statistics allows us to obtain an estimate of the universal factor, ξ 3 D , in three dimensions (3D), in addition to providing a systematic treatment of finite‐ N corrections. We show that in 3D, finite‐ N corrections lead to unphysical values for ξ 3 D , thereby suggesting that a simple DFT applied to a small number of particles may not be suitable in 3D. We then perform an analogous calculation for the two dimensional (2D) system in the infinite‐scattering length regime, and obtain a value of ξ 2 D = 1. Owing to the unique properties of the Thomas‐Fermi energy density‐functional in 2D our result, in contrast to 3D, is exact and therefore requires no finite‐ N corrections. (© 2007 by Astro Ltd., Published exclusively by WILEY‐VCH Verlag GmbH & Co. KGaA)