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

Consider N bosons in a finite box Lambda = [0, L](3) subset of 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 (lim) over bar (rho -> 0) (lim) over bar (L ->infinity, N/L3 ->rho) (e(0)(rho) - 4 pi a rho/(4 pi a)(5/2)(rho)(3/2)) <= 16/15 pi(2), where a is the scattering length of the potential. Previously, an upper bound of the form C16/15 pi(2) for some constant C > 1 was obtained in (Erdos et al. in Phys. Rev. A 78: 053627, 2008). Our result proves the upper bound of the prediction by Lee and Yang (Phys. Rev. 105(3): 1119-1120, 1957) and Lee et al. (Phys. Rev. 106(6): 1135-1145, 1957).