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]³⊂ R³ 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 ᵨ→₀ ₗ → ∞, ₙ/ₗ₃ → ᵨ (e₀(ρ)- 4 πa ρ(4 πa)⁵/²(ρ)³/²)≤ 16/15π², where a is the scattering length of the potential. Previously, an upper bound of the form C 16/15π² for some constant C > 1 was obtained in ESY. Our result proves the upper bound of the the prediction by Lee-Yang LYang and Lee-Huang-Yang LHY.