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

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

2009-03-31

arxiv.org/abs/0903.5347v2

J. Stat. Phys, Volume 136, Issue 3, pp.453-503 2009

Physics

Mathematical Physics

62 pages, no figures

Scientific paper

10.1007/s10955-009-9792-3

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 \[\bar\lim_{\rho\to0} \bar \lim_{L \to \infty, N/L^3 \to \rho} (\frac{e_0(\rho)- 4 \pi a \rho}{(4 \pi a)^{5/2}(\rho)^{3/2}})\leq \frac{16}{15\pi^2}, \] 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 \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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