Third-order term in the dilute Bose gas ground-state energy
In plain words
The energy of a dilute repulsive Bose gas is known rigorously to two terms in powers of the density. The next predicted term, with a logarithm, is proven only from above.
Precise statement
With $\hbar^{2}/(2m) = 1$, the ground-state energy density is predicted to be $e(\rho) = 4 \pi a \rho^2 [1 + (128/(15 \sqrt{\pi})) (\rho a^3)^{1/2} + 8 (4 \pi/3 - \sqrt{3}) \rho a^3 \operatorname{log}(\rho a^3) + o(\rho a^3 \operatorname{log}(\rho a^3))]$ (Wu, 1959). Prove the matching lower bound for general short-range $v \ge 0$. Answer: a proof of the two-sided expansion.
What would settle it
A lower bound on $e(\rho)$ matching the Wu term.
Status in the literature
Unverified note
The two-term Lee-Huang-Yang formula is proven (Yau and Yin 2009, upper bound; Fournais and Solovej 2020, lower bound); a third-order upper bound in the thermodynamic limit was proven by Brooks, Oldenburg, Saint Aubin and Schlein (arXiv 2506.04153, 2025).