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, ${ε}{}4{π}{}a[1+(128∕15√{{π}}){({}{a}³)}1∕2{S}_{{λ}}]+O({{}}²{}ln{{0.2em}{0ex}}{}{})$, as ${}{→}0$, with a constant satisfying $1{}{S}_{{λ}}{}1+C{λ}$. 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 ${S}_{{λ}}=1$ independent of ${λ}$.