8 problems

Uniformity conjecture. The asymptotics above should be uniform in $\mu$, even when $\mu/k_{\rm F}$ tends to infinity. This would extend the high-density ground-state energy formula…

For a Riesz interaction in the Coulomb case $s=d-2$, or for the logarithmic interaction $f(r)=-\log r$ in dimension $d=2$, consider electrons embedded in a uniform background of po…

Wigner crystallization conjecture. The minimum of the jellium energy should be achieved by a crystal, called a Wigner crystal.

Let $d\geqslant 1$ and $-2\leqslant s<d$. Consider the Jellium model in $d$ dimensions with Riesz interaction and define its lowest energy in a background region $\Omega_N$ of volu…

In dimension $3$ and for the Coulomb interaction $s=1$, let $\mathsf C_{\mathrm{UEG}}(1,3)$ denote the Uniform Electron Gas next-order constant and let $\mathsf C_{\mathrm{LO}}$ de…

Let $C_{\mathrm{Jel}}(s,d)$ be the Jellium next-order constant obtained by minimizing over general configurations, and let $C_{\mathrm{Jel}}^{\mathrm{lattice}}(s,d)$ be the corresp…

Let $(x_1^n,\ldots,x_n^n)$ be a sequence of minimizing configurations, and let the additional decay hypothesis imposed in the paper be denoted by the relevant hypothesis on the fie…

In two dimensions, consider the renormalized energy for the Riesz gases with power-law exponents in the range under consideration. Abrikosov's conjecture. The renormalized energy i…