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 \(\mathfrak {a}\) 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 \(4\pi \rho \mathfrak {a}\) and shows that corrections are smaller than \(C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}\), 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 \(\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}\), in agreement with the Lee–Huang–Yang prediction.