Concentration of nuclear charge minimizes the Dirac gap level
Let $\sigma_1, \sigma_2, \sigma_3$ be the Pauli matrices and $$\beta = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}, \qquad \alpha_j = \begin{pmatrix} 0 & \sigma_j \\ \sigma_j & 0 \end{pmatrix} \quad (j = 1, 2, 3).$$ For a nonnegative finite Borel measure $\mu$ on $\mathbb{R}^3$ let $$D_\mu = -i\alpha \cdot \nabla + \beta - V_\mu, \qquad V_\mu(x) = \int_{\mathbb{R}^3} \frac{d\mu(y)}{|x - y|},$$ and define its first min–max level, over upper spinors $\varphi$ and lower spinors $\chi$, by $$\lambda_1(\mu) = \inf_{0 \ne \varphi \in C_c^\infty(\mathbb{R}^3;\mathbb{C}^2)} \ \sup_{\chi \in C_c^\infty(\mathbb{R}^3;\mathbb{C}^2)} \frac{\langle (\varphi, \chi), D_\mu (\varphi, \chi) \rangle}{\|\varphi\|_2^2 + \|\chi\|_2^2}.$$ For $\mu = \nu\delta_0$ with $0 < \nu < 1$ this is $\sqrt{1 - \nu^2}$, the lowest eigenvalue of the distinguished self-adjoint realization of the Dirac–Coulomb operator. For the Schrödinger operator $-\Delta/2 - V_\mu$ the lowest eigenvalue is concave in $\mu$, so at fixed mass it is smallest for a point charge. Esteban, Lewin and Séré showed that when $\mu(\mathbb{R}^3) < 1$ the operator $D_\mu$ has a distinguished self-adjoint extension and that $\lambda_1(\mu)$, whenever it is larger than $-1$, is its lowest eigenvalue in the gap $(-1, 1)$. Since $\lambda_1$ is monotone in $\mu$, Newton's theorem gives the inequality below for radially symmetric $\mu$. For every mass below a critical value $\nu_1 \ge 2/(\pi/2 + 2/\pi)$, they proved that a measure minimizing $\lambda_1$ at fixed mass exists and that every minimizer is concentrated on a compact set of Lebesgue measure zero.
Problem. Is $\lambda_1(\mu) \ge \sqrt{1 - \nu^2}$ for every $\nu \in (0, 1)$ and every nonnegative finite Borel measure $\mu$ on $\mathbb{R}^3$ of mass $\mu(\mathbb{R}^3) = \nu$?
Source: Esteban, Lewin and Séré (2021), Conjecture 1 (optimality for a delta).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, Coulomb potentials, Dirac operators, eigenvalue optimization
References
@article{esteban2021ii,
author = {Esteban, Maria J. and Lewin, Mathieu and S{\'e}r{\'e}, {\'E}ric},
title = {Dirac--{C}oulomb operators with general charge distribution {II}. {T}he lowest eigenvalue},
journal = {Proc. Lond. Math. Soc. (3)},
volume = {123},
number = {4},
year = {2021},
pages = {345--383},
doi = {10.1112/plms.12396},
eprint = {2003.04051},
archivePrefix = {arXiv},
}
@article{esteban2021i,
author = {Esteban, Maria J. and Lewin, Mathieu and S{\'e}r{\'e}, {\'E}ric},
title = {Dirac--{C}oulomb operators with general charge distribution {I}. {D}istinguished extension and min-max formulas},
journal = {Ann. H. Lebesgue},
volume = {4},
year = {2021},
pages = {1421--1456},
doi = {10.5802/ahl.106},
eprint = {2003.04004},
archivePrefix = {arXiv},
}
@incollection{esteban2023,
author = {Esteban, Maria J. and Lewin, Mathieu and S{\'e}r{\'e}, {\'E}ric},
title = {Which nuclear shape generates the strongest attraction on a relativistic electron? {A}n open problem in relativistic quantum mechanics},
booktitle = {Mathematics Going Forward: Collected Mathematical Brushstrokes},
editor = {Morel, Jean-Michel and Teissier, Bernard},
series = {Lecture Notes in Math.},
volume = {2313},
publisher = {Springer},
address = {Cham},
year = {2023},
pages = {487--497},
doi = {10.1007/978-3-031-12244-6_34},
}
@misc{aim-dirac-charge-concentration,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Concentration of nuclear charge minimizes the Dirac gap level}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/101-dirac-charge-concentration.md},
note = {Entry 101 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-00341, "Concentration of nuclear charge minimizes the Dirac gap level", clopen.solutions.