Moments of the ground state density for the d-dimensional Fermi gas in an harmonic trap

Abstract

We consider properties of the ground state density for the d-dimensional Fermi gas in an harmonic trap. Previous work has shown that the d-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the 2kth non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Rek > -d/2. The moments, and the smoothed density, permit expansions in 1/M2, where M = M + (d + 1)/2, with M denoting the s..