A second order upper bound on the ground state energy of a Bose gas beyond the Gross-Pitaevskii regime

We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N^{−1+κ}, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.