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

Consider $N$ bosons in a finite box $\Lambda= [0,L]^3\subset \bR^3$ interacting via a two-body nonnegative soft potential $V= \lambda \tilde V$ with $\tilde V$ fixed and $\lambda>0$ small. We will take the limit $L, N \to \infty$ by keeping the density $\varrho= N/L^{3}$ fixed and small. We construct a variational state which gives an upper bound on the ground state energy per particle $\e$ $$ \e \le 4\pi\varrho a \Big [1+ \frac{128}{15\sqrt{\pi}}(\varrho a^3)^{1/2}S_\lambda \Big ] + O(\varrho^2|\log\varrho|), \quad {as $\varrho\to 0$} $$ with a constant satisfying $$ 1\leq S_\lambda \leq 1+C\lambda. $$ Here $a$ is the scattering length of $V$ and thus depends on $\lambda$. In comparison, the prediction by Lee-Yang \cite{LYang} and Lee-Huang-Yang \cite{LHY} asserts that $S_\lambda=1$ independent of $\lambda$.