The Lee-Huang-Yang energy for a dilute gas of hard spheres: an upper bound

We consider a quantum gas consisting of $N$ hard spheres with radius $\mathfrak{a}>0$, obeying bosonic statistics and moving in the box $\mathrm{\Lambda }=[0;L{]}^3$ with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with $N,L\to \mathrm{\infty }$ at fixed density $\rho =N\mathrm{/}{L}^3$. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit $\rho {\mathfrak{a}}^3\ll 1$.