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 4 π ρ a and shows that corrections are smaller than C ρ a ( ρ a 3 ) 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 ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee-Huang-Yang prediction.