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Accepted · first revision · record

Title: Sharp Lieb-Oxford constant for the indirect Coulomb energy
Plain: The Lieb-Oxford inequality bounds how far the electrostatic energy of many charges can fall below the value for a smeared-out charge cloud, using only their density. The best constant lies between 1.44 and 1.58 and is unknown.
Precise: For every $N$-particle probability density on $R^{3N}$ with one-body density $\rho$, $\left\langle \sum_{i<j} e^2/\mid x_i - x_j \mid \right\rangle - (e^2/2) \int \int \rho(x) \rho(y)/\mid x - y \mid\, dx\, dy \ge -C_{\mathrm{LO}} e^2 \int \rho^{4/3}\, dx$. Determine the optimal $C_{\mathrm{LO}}$, known to satisfy $1.4442 \le C_{\mathrm{LO}} \le 1.58$, and decide whether it equals the uniform-electron-gas constant, the low-density limit of the jellium energy per particle in units $e^2 \rho^{1/3}$. That constant is $\ge 1.4442$, the bcc Wigner-crystal value, with equality if mathph.crystallization.jellium-bcc holds.
Settled by: A proof of the inequality with the conjectured constant, or a trial state whose ratio exceeds the uniform-electron-gas value.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: open
Status note: Lewin, Lieb, Seiringer (Lett. Math. Phys. 2022) lowered the upper bound from 1.64 to 1.58 and recorded the lower bound 1.4442.
Posed since: (empty)