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

Consider $N$ bosons in a finite box $\Lambda ={[0,L]}^3\subset {\mathbb{R}}^3$ interacting via a two-body non-negative 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 $\epsilon$, $\epsilon ⩽4\pi \varrho a[1+(128∕15\sqrt{\pi }){\left(\varrho a^3\right)}^{1∕2}{S}_{\lambda }]+O\left({\varrho }^2\mid \ln \varrho \mid \right)$, as $\varrho \to 0$, 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 $\lambda$.