Equality of the Jellium and the Uniform Electron Gas Energies

Jellium, a system of electrons in a uniform neutralizing background, and the uniform electron gas (UEG), a system of electrons with constant density and no background, are two standard models of the electron gas. We prove that the ground-state energy per unit volume of Jellium, as defined by Lieb and Narnhofer, coincides with the UEG energy defined by Lewin, Lieb and Seiringer, i.e. $$e_{\mathrm{J}\mathrm{e}\mathrm{l}}^{\mathrm{\hslash }}({\rho }_0)=e_{\mathrm{U}\mathrm{E}\mathrm{G}}^{\mathrm{\hslash }}({\rho }_0)\mathrm{.}$$ This equality was previously established in the classical case by Lewin, Lieb and Seiringer and independently by Cotar and Petrache.