Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions

We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive analytically a differential equation connecting these densities, which so far have been proven only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is identified uniquely. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation $\tau_{TF}[\rho]$ between kinetic and particle density is fulfilled also for the asymptotic quantum densities $\tau(r)$ and $\rho(r)$ including their leading-order oscillating terms.