The concentration conjecture for Dirac--Coulomb operators
Abstract
We prove the concentration conjecture for the three-dimensional Dirac operator with mass m>0 and an attractive Coulomb potential generated by finitely many point charges. For a fixed total charge G∈(0,1], the lowest eigenvalue in the spectral gap is at least m√(1−G2), with equality precisely when all the charge is concentrated at one point. The critical one-center operator is understood as its distinguished extension. The result also extends to arbitrary nonnegative finite charge measures. The proof uses a strict two-center estimate for the weighted free Dirac resolvent and a Birman-Schwinger comparison. We establish the two-center estimate through an auxiliary axial operator, combining a zero-energy bound and a scalar resolvent estimate with a radial decomposition, an angular Schur complement, and exact polynomial inequalities.
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BibTeX
@misc{ammari2026concentration,
title = {{The concentration conjecture for Dirac--Coulomb operators}},
author = {Habib Ammari and Jiayu Qiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28193},
eprint = {2610.28193},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28193},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The concentration conjecture for Dirac--Coulomb operators
AU - Habib Ammari
AU - Jiayu Qiu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28193
UR - https://arxiv.org/abs/2610.28193
AB - We prove the concentration conjecture for the three-dimensional Dirac operator with mass m>0 and an attractive Coulomb potential generated by finitely many point charges. For a fixed total charge G∈(0,1], the lowest eigenvalue in the spectral gap is at least m√(1−G^2), with equality precisely when all the charge is concentrated at one point. The critical one-center operator is understood as its distinguished extension. The result also extends to arbitrary nonnegative finite charge measures. The proof uses a strict two-center estimate for the weighted free Dirac resolvent and a Birman-Schwinger comparison. We establish the two-center estimate through an auxiliary axial operator, combining a zero-energy bound and a scalar resolvent estimate with a radial decomposition, an angular Schur complement, and exact polynomial inequalities.
ER -