Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion

We prove an upper bound on the energy density of the dilute spin-\(\frac {1}{2}\) Fermi gas capturing the leading correction to the kinetic energy\(8\pi a\rho _\uparrow\rho _\downarrow\) with an error of size smaller than\(a\rho^{2}(a^ 3\rho)^{1/3-\varepsilon}\) for any\(\varepsilon> 0\), where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).