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 [Formula: see text]-dimensional Fermi gas in an harmonic trap. Previous work has shown that the [Formula: see text]-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 [Formula: see text]th non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Re[Formula: see text]. The moments, and the smoothed density, permit expansions in [Formula: see text], where [Formula: see text], with [Formula: see text] denoting the shell label. The moment expansion substituted in the second-order recurrence gives a generalization of the Harer–Zagier recurrence, satisfied by the coefficients of the [Formula: see text] expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.