The ionization conjecture

Experiments show that a neutral atom binds at most one or two extra electrons. The ionization conjecture asks for a proof, from the many-electron Schrödinger equation, of a weaker statement: the number of extra electrons a nucleus can bind stays bounded however large its charge. A proof claimed in September 2026 has not yet been independently checked.

Problem

For a fixed point nucleus of charge $Z > 0$ and $N$ electrons, the Hamiltonian in Simon's units is

$$H(N,Z) = \sum_{i=1}^{N} \Big( -\Delta_i - \frac{Z}{|x_i|} \Big) + \sum_{1 \le i < j \le N} \frac{1}{|x_i - x_j|}.$$

It acts on $\bigwedge^N L^2(\mathbb{R}^3; \mathbb{C}^2)$, the wave functions antisymmetric in the joint space and spin variables of the electrons. Let $E(N,Z) = \inf \operatorname{spec} H(N,Z)$ and $E(0,Z) = 0$. By the HVZ theorem $E(N,Z) \le E(N-1,Z)$, and the nucleus binds $N$ electrons when the inequality is strict. Let $N_c(Z)$ be the largest such $N$ (Simon's $N_0(Z)$).

Conjecture (Problem 9). There is a constant $C$ with $N_c(Z) \le Z + C$ for all $Z$.

Simon adds that $N_c(Z)$ may always be $Z$ or $Z + 1$.

Background

This is Problem 9 on Barry Simon's 2000 list. He first asked it in 1984, in a note added in proof to his first list, right after Lieb, Sigal, Simon and Thirring proved $N_c(Z)/Z \to 1$. The heuristic is electrostatic: a far-away electron is attracted only while the net charge $Z - (N-1)$ of the rest is positive. A proof must control many-body correlations and use the Pauli principle. For bosonic "electrons" the conjecture fails: $\liminf_{Z \to \infty} N_c(Z)/Z \ge t_c > 1$ (Benguria and Lieb), and in fact $N_c(Z)/Z \to t_c \approx 1.21$ (Solovej) [6].

What is known

• $N_c(Z)$ is finite (Ruskai and Sigal, 1982, independently) and at least $Z$ (Zhislin, 1960) [1, 6].
• Non-asymptotic bounds: $N_c(Z) < 2Z + 1$ for all $Z$ (Lieb, 1984), which settles hydrogen [2, 6]. Later, $N_c(Z) < 1.22\,Z + 3Z^{1/3}$ (Nam, 2012) [4], and $N_c(Z) < 1.1185\,Z + O(Z^{1/3})$ (Hundertmark, Pattakos and Schulz, 2025 preprint) [5].
• Asymptotics: $N_c(Z)/Z \to 1$ (Lieb, Sigal, Simon and Thirring, 1984). In 1990 Fefferman–Seco and Seco–Sigal–Solovej proved $N_c(Z) \le Z + O(Z^{5/7})$, still the best asymptotic bound in the 2022 survey [6].
• The conjecture holds in Hartree–Fock theory (Solovej, 2003) [3], and in Thomas–Fermi–Dirac–von Weizsäcker and Müller theory (Frank, Nam and Van Den Bosch, 2018) [6].
• Claimed resolution. On 24 September 2026 OpenAI released a manuscript, Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms. It reports a proof, in the full model above, that $M$ fixed nuclei with charges at least 1 and total charge $Z$ bind at most $Z + CM$ electrons, with $C$ universal. For one nucleus this is the conjecture. The manuscript is unrefereed and has not been independently verified, and this bound is not formalized: the Lean formalizations released with it cover companion results on removing many electrons, and a third-party Lean development (October 2026) covers only the manuscript's bounds on the first ionization energy and the outer radius of neutral atoms.

What would count as progress

• An independent check of the 2026 manuscript: a verified proof, a gap in one of its steps, or a Lean formalization of $N_c(Z) \le Z + C$.
• A smaller asymptotic exponent: $N_c(Z) \le Z + O(Z^{a})$ with $a < 5/7$.
• A smaller leading constant: $N_c(Z) \le bZ + O(Z^{1/3})$ with $b < 1.1185$.

References

1. B. Simon, "Schrödinger operators in the twenty-first century", in Mathematical Physics 2000, Imperial College Press, 2000, pp. 283–288. doi:10.1142/9781848160224_0014
2. E. H. Lieb, "Bound on the maximum negative ionization of atoms and molecules", Phys. Rev. A 29 (1984), 3018–3028. doi:10.1103/PhysRevA.29.3018
3. J. P. Solovej, "The ionization conjecture in Hartree–Fock theory", Ann. of Math. 158 (2003), 509–576. doi:10.4007/annals.2003.158.509
4. P. T. Nam, "New bounds on the maximum ionization of atoms", Comm. Math. Phys. 312 (2012), 427–445. doi:10.1007/s00220-012-1479-y
5. D. Hundertmark, N. Pattakos and M. R. Schulz, "On the excess charge problem of atoms", preprint, 2025. arXiv:2504.18487
6. P. T. Nam, "The ionization problem in quantum mechanics", survey, 2022. arXiv:2206.15393

Source

Imported from Simon's problems in mathematical physics 2000-9, curated by Barry Simon. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.