, Volume 136, Issue 3, pp 453-503

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

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Consider N bosons in a finite box Λ=[0,L]3R 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 $$\overline{\lim}_{\rho\to0}\overline{\lim}_{L\to\infty,\,N/L^3\to \rho}\biggl(\frac{e_0(\rho)-4\pi a\rho}{(4\pi a)^{5/2}(\rho)^{3/2}}\biggr )\leq\frac{16}{15\pi^2},$$ where a is the scattering length of the potential. Previously, an upper bound of the form C16/15π 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).

Partially supported by NSF grants DMS-0757425, 0804279.