Uniform boundedness of the first atomic ionization energy
For $Z > 0$ and an integer $N \ge 1$, let $E(N,Z)$ be the bottom of the spectrum of $$H_{N,Z} = \sum_{j=1}^N \left( -\frac12 \Delta_{x_j} - \frac{Z}{|x_j|} \right) + \sum_{1 \le i < j \le N} \frac{1}{|x_i - x_j|}$$ on the antisymmetric space $\bigwedge^N L^2(\mathbb{R}^3; \mathbb{C}^2)$, defined through its quadratic form, and set $E(0,Z) = 0$. The nucleus has infinite mass and each electron has exactly two spin states. For an integer $Z \ge 1$, the first ionization energy of the neutral atom is $$I_1(Z) = E(Z-1, Z) - E(Z, Z).$$ It is a difference of two energies of order $Z^{7/3}$, and the known large-$Z$ expansion of the ground-state energy, to order $Z^{5/3}$, is far from the accuracy needed to control it. In Hartree–Fock theory, Solovej proved that the ionization energy of an atom is bounded independently of the nuclear charge.
Problem. Is $\sup_{Z \in \mathbb{N}} I_1(Z) < \infty$?
Source: The ionization conjecture, which Solovej (2016) traces to Simon (1984) and Lieb and Seiringer (2010); for the first ionization energy in the form of Solovej (2016), Section 2. Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Coulomb systems, ionization conjecture, many-body Schrödinger operators
References
@article{solovej2016,
author = {Solovej, Jan Philip},
title = {A new look at {T}homas--{F}ermi theory},
journal = {Mol. Phys.},
volume = {114},
number = {7-8},
year = {2016},
pages = {1036--1040},
doi = {10.1080/00268976.2015.1130273},
eprint = {1601.00497},
archivePrefix = {arXiv},
}
@article{solovej2003,
author = {Solovej, Jan Philip},
title = {The ionization conjecture in {H}artree--{F}ock theory},
journal = {Ann. of Math. (2)},
volume = {158},
number = {2},
year = {2003},
pages = {509--576},
doi = {10.4007/annals.2003.158.509},
eprint = {math-ph/0012026},
archivePrefix = {arXiv},
}
@incollection{simon1984,
author = {Simon, Barry},
title = {Fifteen problems in mathematical physics},
booktitle = {Perspectives in Mathematics},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {1984},
pages = {423--454},
}
@book{lieb2010,
author = {Lieb, Elliott H. and Seiringer, Robert},
title = {The Stability of Matter in Quantum Mechanics},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2010},
}
@misc{aim-uniform-first-ionization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniform boundedness of the first atomic ionization energy}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/099-uniform-first-ionization.md},
note = {Entry 099 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that a molecule with $M$ fixed nuclei of real charges at least one and total nuclear charge $Z$ strictly binds at most $Z+CM$ electrons, where $C$ is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge $Z\ge1$, the first ionization energy …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00732, "Uniform boundedness of the first atomic ionization energy", clopen.solutions.