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Extended Thomas-Fermi Density Functional for the Unitary Fermi Gas

We determine the energy density $ξ(3/5) n ε_F$ and the gradient correction $λ\hbar^2(\nabla n)^2/(8m n)$ of the extended Thomas-Fermi (ETF) density functional, where $n$ is number density and $ε_F$ is Fermi energy, for a trapped two-components Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. {\bf 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 $γN^{1/9} \hbar ω$ 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.