Extended Thomas-Fermi density functional for the unitary Fermi gas

We determine the energy density ξ(3∕5)nεF and the gradient correction λ²(∇n)²∕(8mn) of the extended Thomas-Fermi (ETF) density functional, where n is the number density and εF is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that ξ=0.455 and λ=0.13 give the best fit of the DMC data with an even number N of particles. We also study the odd-even splitting γN1∕9ω of the ground-state energy for the unitary gas in a harmonic trap of frequency ω determining the constant γ. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.