Semiclassical mean-field limit of the Levy-Lieb functional

Let $d\geqslant 1$, and let $\mathcal{E}_N(f)$ and $\mathcal{E}^{\rm TF}(f)$ be the Levy-Lieb and Thomas–Fermi density functionals, respectively. The parameters satisfy the semiclassical mean-field scaling

$$N\to\infty,\qquad hN^{1/d}\to 1,\qquad \lambda N\to 1.$$

Semiclassical mean-field limit conjecture. For every function $f$ satisfying $f\geqslant 0$, $\sqrt{f}\in H^1(\mathbb{R}^d)$ and $\int_{\mathbb{R}^d}f=1$, one has

$$\mathcal{E}_N(f)\to \mathcal{E}^{\rm TF}(f).$$

The conjecture predicts convergence of the Levy-Lieb functional to the Thomas–Fermi functional for a broad class of potentials in the semiclassical mean-field regime. The stated potential assumptions include Coulomb interactions in $\mathbb{R}^3$; the condition $\sqrt f\in H^1(\mathbb{R}^d)$ is necessary for finiteness of the kinetic contribution. The source presents this as an open problem.