Convexity of atomic ground-state energy in electron number
For $Z > 0$ and an integer $N \ge 1$, let $E(N,Z)$ be the bottom of the spectrum of the Hamiltonian, in atomic units, of $N$ electrons bound to one fixed point nucleus of charge $Z$, $$H_{N,Z} = \sum_{j=1}^{N} \Bigl( -\tfrac12 \Delta_{x_j} - \frac{Z}{|x_j|} \Bigr) + \sum_{1 \le i < j \le N} \frac{1}{|x_i - x_j|},$$ acting on the antisymmetric space $\bigwedge^N L^2(\mathbb{R}^3; \mathbb{C}^2)$ and defined through its quadratic form, and set $E(0,Z) = 0$. The inequality below holds for $N = 1$, since dropping the electron repulsion gives $E(2,Z) \ge 2E(1,Z)$, and it holds for non-interacting electrons. Lieb asked whether the ground-state energy is convex in the number of electrons for every external potential. Di Marino, Lewin and Nenna showed that it is not: for three electrons in the field of six far-apart nuclei with small non-integer charges, the energy exceeds the average of the energies for two and for four electrons (2024).
Problem. Is $$E(N+1,Z) + E(N-1,Z) \ge 2E(N,Z)$$ for every integer nuclear charge $Z \ge 1$ and every integer $N \ge 1$?
Source: Simon (1984), Problem 10A, for atoms; Lieb (1983), Question 7, asked it for every external potential. See Lewin (2025), Open Problem 2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, ionization energies, many-electron atoms
References
@incollection{simon1984,
author = {Simon, Barry},
title = {Fifteen problems in mathematical physics},
booktitle = {Perspectives in Mathematics: Anniversary of {O}berwolfach 1984},
editor = {J{\"a}ger, Willi and Moser, J{\"u}rgen and Remmert, Reinhold},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {1984},
pages = {423--454},
}
@article{lieb1983,
author = {Lieb, Elliott H.},
title = {Density functionals for {C}oulomb systems},
journal = {Int. J. Quantum Chem.},
volume = {24},
number = {3},
year = {1983},
pages = {243--277},
doi = {10.1002/qua.560240302},
}
@article{lewin2025,
author = {Lewin, Mathieu},
title = {Some open mathematical problems concerning charged quantum particles},
journal = {C. R. Phys.},
volume = {26},
year = {2025},
pages = {369--380},
doi = {10.5802/crphys.249},
}
@article{dimarino2024,
author = {Di Marino, Simone and Lewin, Mathieu and Nenna, Luca},
title = {Ground state energy is not always convex in the number of electrons},
journal = {J. Phys. Chem. A},
volume = {128},
number = {49},
year = {2024},
pages = {10697--10706},
doi = {10.1021/acs.jpca.4c06345},
eprint = {2409.08632},
archivePrefix = {arXiv},
}
@misc{aim-atomic-energy-convexity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Convexity of atomic ground-state energy in electron number}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/100-atomic-energy-convexity.md},
note = {Entry 100 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00329, "Convexity of atomic ground-state energy in electron number", clopen.solutions.