In recent years there has been substantial progress in the mathematical understanding of the low-energy properties of dilute Bose gases in the thermodynamic limit. In particular the validity of a celebrated second order asymptotics of the ground state energy of dilute bosons – predicted by Lee, Huang and Yang in 1957 – has been fully established in the case of integrable (non-negative) interactions. Still, the derivation of an upper bound in agreement with this asymptotics for hard sphere bosons remains an open problem.

In this talk we discuss how a simple trial state introduced by Bijl-Dingle-Jastrow back in the 50s can be used to derive an upper bound for the ground state energy of a dilute Bose gas of hard sphere compatible with the Lee-Huang-Yang expansion, up to errors of the order of the sub-leading correction. Our proof is based on the identification of precise cancellations occurring in the computation of the energy of the Bijl-Dingle-Jastrow trial state, which can also be inspected via cluster expansion methods.

Joint work with G. Basti, A. Giuliani, A. Olgiati, G. Pasqualetti, B. Schlein