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

Consider N bosons in a finite box Lambda=[0,L](3)subset of R-3 interacting via a two-body non-negative soft potential V=lambda V with V fixed and lambda>0 small. We will take the limit L,N ->infinity by keeping the density rho=N/L-3 fixed and small. We construct a variational state, which gives an upper bound on the ground-state energy per particle epsilon, epsilon <= 4 pi rho a[1+(128/15 pi)(rho a(3))S-1/2(lambda)]+O(rho(2)parallel to ln rho parallel to), as rho -> 0, with a constant satisfying 1 <= S-lambda <= 1+C lambda. Here a is the scattering length of V and thus depends on lambda. 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 S-lambda=1 independent of lambda.