Body-centered cubic ground state of classical 3D jellium
In plain words
Point charges in a uniform background of opposite charge are expected to form a body-centered cubic crystal, the Wigner crystal. No proof exists that this arrangement has the lowest energy.
Precise statement
For $N$ point charges $e$ in a neutralizing uniform background of density $\rho0$ in $R^{3}$ (Coulomb $e^{2}/r$), prove that in the thermodynamic limit the minimal energy per particle equals the bcc value, about $-0.8959\,e^{2}/r_{s}$ with $r_{s}=(3/(4\pi \rho0))^{1/3}$, and that minimizers are bcc up to $o(N)$ particles. The quantum version asks the same for the electron gas as $r_{s}\to\infty$.
What would settle it
A proof that no configuration beats the bcc Madelung energy per particle in the thermodynamic limit.
Status in the literature
Open in both the classical and the quantum low-density version.