MATHPH
In the literature: open
Uniform bound on the ionization energy of neutral atoms
In plain words
The energy needed to remove one electron from a neutral atom is a few eV for every element in the periodic table. Prove that it stays bounded as Z grows.
Precise statement
Let $I(Z) = E(Z-1,Z) - E(Z,Z)$ for the Hamiltonian $H_{N,Z}$ of mathph.ionization.excess-charge-bound. Prove $\operatorname{sup}_Z I(Z) < \infty$, i.e. $I(Z) \le C m e^4/\hbar^2$ with $C$ independent of $Z$, or show $I(Z)$ is unbounded.
What would settle it
A Z-independent upper bound on E(Z-1,Z) - E(Z,Z) derived for the full many-electron Hamiltonian.
Status in the literature
Proved in Hartree-Fock theory (Solovej 2003); for the Schrodinger Hamiltonian the best known bound grows with $Z, I(Z) \le C Z^{5/7}$ (Seco, Sigal, Solovej 1990).