The ground state energy of a low density Bose gas: a second order upper bound

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Consider N bosons in a finite box Λ= [0,L]³⊂ ³ interacting via a two-body nonnegative soft potential V= λ V with V fixed and λ>0 small. We will take the limit L, N → ∞ by keeping the density = N/L³ fixed and small. We construct a variational state which gives an upper bound on the ground state energy per particle ≤ 4π a [1+ 128/15√π(a³)¹/²Sλ] + O(²||), as → 0 with a constant satisfying 1≤ Sλ≤ 1+Cλ. Here a is the scattering length of V and thus depends on λ. In comparison, the prediction by Lee-Yang LYang and Lee-Huang-Yang LHY asserts that Sλ=1 independent of λ.