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Energetics of a strongly correlated Fermi gas

The energy of the two-component Fermi gas with the s-wave contact interaction is a simple linear functional of its momentum distribution: $$E_\text{internal}=\hbar^2ΩC/4πam+\sum_{\vect kσ}(\hbar^2 k^2/2m)(n_{\vect kσ}-C/k^4)$$ where the external potential energy is not included, $a$ is the scattering length, $Ω$ is the volume, $n_{\vect kσ}$ is the average number of fermions with wave vector $\vect k$ and spin $σ$, and $C\equiv\lim_{\vect k\to\infty} k^4 n_{\vect k\up} =\lim_{\vect k\to\infty} k^4 n_{\vect k\down}$. This result is a \textit{universal identity}. Its proof is facilitated by a novel mathematical idea, which might be of utility in dealing with ultraviolet divergences in quantum field theories. Other properties of this Fermi system, including the short-range structure of the one-body reduced density matrix and the pair correlation function, and the dimer-fermion scattering length, are also studied.