The Second Order Upper Bound for the Ground Energy of a Bose Gas

Consider $N$ bosons in a finite box $\mathrm{\Lambda }=[0,L{]}^3\subset {\mathbf{R}}^3$ interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle $${\stackrel{}{\lim }}_{\rho \to 0}{\stackrel{}{\lim }}_{L\to \mathrm{\infty },N\mathrm{/}{L}^3\to \rho }(\frac{e_0(\rho )-4\pi a\rho }{(4\pi a{)}^{5\mathrm{/}2}(\rho {)}^{3\mathrm{/}2}})\le \frac{16}{15{\pi }^2},$$ where $a$ is the scattering length of the potential. Previously, an upper bound of the form $C16\mathrm{/}15{\pi }^2$ for some constant $C>1$ was obtained in \cite{ESY}. Our result proves the upper bound of the the prediction by Lee-Yang \cite{LYang} and Lee-Huang-Yang \cite{LHY}.