Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

We consider a gas of bosons interacting through a hard-sphere potential with radius a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa and shows that corrections are smaller than Cρa(ρa3)1/2, for a sufficiently large constant C > 0. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order ρa(ρa3)1/2, in agreement with the Lee–Huang–Yang prediction.