5 problems

Thomas–Fermi dynamical neutrality conjecture. For every initial state $(\varphi_0,\rho_0)$ satisfying $0\leqslant\rho_0\in L^1(\mathbb{R}^3)\cap L^{5/3}(\mathbb{R}^3)$ and…

Let $z_1=z_2=z/2$ be the nuclear charges of a homonuclear diatomic molecule, let $R=|R_1-R_2|$ be the internuclear distance, and let $D^{\mathrm{GS}}(\underline{Z},\underline{R})$…

Let $d\in\{2,3\}$ and let $\rho\in L^1(\mathbb{R}^d,\mathbb{R}_+)$ with $\sqrt{\rho}\in H^1(\mathbb{R}^d)$. Define the lowest kinetic energy at fixed density by … Let $c_{\rm TF}$…

Finer-scale density approximation conjecture. The estimate above can be improved to show that $\varrho^{\mathrm{af}}$ is close to $\varrho^{\mathrm{TF}}_\beta$ on finer length scal…

Generalized ionization conjecture for molecules. The large-$Z$ upper and lower limits of the homonuclear diatomic Born–Oppenheimer curve satisfy