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 $\frak{a}$ in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4\pi \rho \frak{a}$ and shows that corrections are smaller than $C \rho \frak{a} (\rho \frak{a}^3)^{1/2}$, 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 is, in fact, of the order $\rho \frak{a} (\rho \frak{a}^3)^{1/2}$, in agreement with the Lee-Huang-Yang prediction.