The second order upper bound for the ground energy of a Bose gas

Horng-Tzer Yau Harvard University Jun Yin University of Wisconsin-Madison

Mathematical Physics mathscidoc:1608.22014

Journal of Statistical Physics, 136, (3), 453–503, 2009
Consider $N$ bosons in a finite box  $ \Lambda = [0,L]^3 \subset 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 $$ \bar{lim|_{\rho \to 0} \bar{lim|_{L \to \infty, N/L^3 \to \rho} ( \ frac{e_0 (\rho) - 4\pi a \rho }{ (4\pi a)^{5/2} (\rho)^{3/2} } ) \le \frac{16}{15 \pi ^2} , $$ 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 (Erdös 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).
Bose gas · Bogoliubov transformation · Variational principle
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@inproceedings{horng-tzer2009the,
  title={The second order upper bound for the ground energy of a Bose gas},
  author={Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824000052404273424},
  booktitle={Journal of Statistical Physics},
  volume={136},
  number={3},
  pages={453–503},
  year={2009},
}
Horng-Tzer Yau, and Jun Yin. The second order upper bound for the ground energy of a Bose gas. 2009. Vol. 136. In Journal of Statistical Physics. pp.453–503. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824000052404273424.
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