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/M̃2, where M̃ = M + (d + 1)/2, with M 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 1/N2 expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.

中文翻译：

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谐波陷阱中 d 维费米气体的基态密度矩

我们考虑基态密度的性质d谐波陷阱中的维费米气体。以前的工作表明，d维傅里叶变换具有非常简单的函数形式。结果表明，这一事实可用于推断密度本身满足三阶线性微分方程，该方程以前在文献中已知，但从其他考虑。它也显示了这如何暗示一个封闭形式的表达式2ķth 密度的非负整数矩，以及二阶递归。两者都可以扩展到一般 Reķ > -d/2. 矩和平滑的密度允许在1/米̃2， 在哪里米̃ = 米 + (d + 1)/2， 和米表示外壳标签。在二阶递归中替换的矩展开给出了 Harer-Zagier 递归的推广，满足1/ñ2随机矩阵理论中高斯酉系综谱密度矩的扩展。