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

Abstract

Consider $N$ bosons in a finite box $Λ= [0,L]^3\subset \mathbf R^3$ 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 \[\bar\lim_{ρ\to0} \bar \lim_{L \to \infty, N/L^3 \to ρ} (\frac{e_0(ρ)- 4 πa ρ}{(4 πa)^{5/2}(ρ)^{3/2}})\leq \frac{16}{15π^2}, \] where $a$ is the scattering length of the potential. Previously, an upper bound of the form $C 16/15π^2$ 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}.

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