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The concentration conjecture for Dirac--Coulomb operators

by H. Ammari and J. Qiu

(Report number 2026-39)

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 $G\in(0,1]$, the lowest eigenvalue in the spectral gap is at least $m\sqrt{1-G^2}$, 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.

Keywords: Dirac-Coulomb operator; concentration conjecture; Birman-Schwinger.

@Techreport{AQ26_1194,
  author = {H. Ammari and J. Qiu},
  title = {The concentration conjecture for Dirac--Coulomb operators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-39.pdf },
  year = {2026}
}

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