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

Abstract

Consider \(N\) bosons in a finite box \(\Lambda= [0,L]^3\subset \mathbf R^3\) 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 \[\overline\lim_{\rho\to0} \overline\lim_{L \to \infty, N/L^3 \to \rho} \left(\frac{e_0(\rho)- 4 \pi a \rho}{(4 \pi a)^{5/2}(\rho)^{3/2}}\right)\leq \frac{16}{15\pi^2}, \] where \(a\) is the scattering length of the potential. Previously, an upper bound of the form \(C 16/15\pi^2\) for some constant \(C > 1\) was obtained in. Our result proves the upper bound of the the prediction by Lee-Yang and Lee-Huang-Yang.