Ground-state energy of a low-density Bose gas: A second-order upper bound

Consider N bosons in a finite box Λ= [0,L]3⊂ R3 interacting via a two-body non-negative 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 ε, ε≤4πa [1+ (128/15π) (a3) 1/2 Sλ] +O (2 ln ), 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 and Yang [Phys. Rev. 105, 1119 (1957)] and Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957)] asserts that Sλ =1 independent of λ.