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

TL;DR

This paper derives a differential equation and explicit formulas for the moments of the ground state density of a $d$-dimensional Fermi gas in a harmonic trap, connecting it to random matrix theory and providing a new analytical framework.

Contribution

It introduces a differential equation approach to the density, derives closed-form moments, and links the results to the Harer--Zagier recurrence in random matrix theory.

Findings

Density satisfies a third order linear differential equation.

Explicit formulas and recurrence relations for moments are obtained.

Connections to the Gaussian unitary ensemble are established.

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 $2k$-th non-negative integer moments of the density, and a second order recurrence. Both can be extended to general Re$\, k > -d/2$. The moments, and the smoothed density, permit expansions in $1/\tilde{M}^2$, where $\tilde{M} = M + (d+1)/2$, with $M$ denoting the shell label. The moment expansion substituted in the second order recurrence gives a generalisation of the Harer--Zagier recurrence, satisfied by the coefficients of the $1/N^2$ expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.