Ionization conjecture for atoms

The question is how many electrons a nucleus of charge $Z$ can bind. Physically atoms bind at most one or two extra electrons, so one expects $N_c(Z)-Z$ to stay bounded. Proving this from the many-body Schrödinger operator $H=\sum_i(-\Delta_i-Z/|x_i|)+\sum_{i<j}1/|x_i-x_j|$ is the ionization conjecture.

What is known: Zhislin proved that a neutral or positive net charge always binds ($N_c(Z)\ge Z$). On the upper side, Ruskai and Sigal showed $N_c(Z)$ is finite; Lieb's elegant argument gives $N_c(Z)<2Z+1$, and asymptotic results (Lieb–Sigal–Simon–Thirring, Fefferman–Seco) show $N_c(Z)/Z\to1$ as $Z\to\infty$. Nam later improved the bound to roughly $N_c(Z)\le 1.22\,Z + 3Z^{1/3}$.

The conjectured sharp form $N_c(Z)\le Z+C$ has been proved in mean-field surrogates: Solovej established it for Hartree–Fock theory, and analogous statements hold in Thomas–Fermi–type and reduced density-matrix models. For the full many-electron Schrödinger operator it remains open; the obstruction is controlling electron–electron screening rigorously without a self-consistent mean field.

Preprint progress (2025): Hundertmark, Pattakos and Schulz report $N_c(Z)<1.1185Z+O(Z^{1/3})$. This improves an explicit finite-$Z$ estimate, not the already known asymptotic coefficient one and not the uniform excess-charge conjecture. Uniform ionization-energy and radius bounds are related but separate targets.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: An improved explicit ionization bound, not a proof of uniformly bounded excess charge. Evidence