Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue

• Published in: arXiv.org
• Main Authors: Esteban, Maria J, Lewin, Mathieu, Séré, Éric
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 11.12.2020
• Subjects: Charge distribution; Coulomb potential; Eigenvalues; Mathematics - Analysis of PDEs; Mathematics - Mathematical Physics; Mathematics - Spectral Theory; Operators; Physics - Mathematical Physics
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2003.04051

Consider the Coulomb potential \(-\mu\ast|x|^{-1}\) generated by a non-negative finite measure \(\mu\). It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator \(-\Delta/2-\mu\ast|x|^{-1}\) is minimized, at fixed mass \(\mu(\mathbb{R}^3)=\nu\), when \(\mu\) is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator \(-i\alpha\cdot\nabla+\beta-\mu\ast|x|^{-1}\). In a previous work on the subject we proved that this operator is self-adjoint when \(\mu\) has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass \(\nu_1\), below which the lowest eigenvalue does not dive into the lower continuum spectrum for all \(\mu\geq0\) with \(\mu(\mathbb{R}^3)<\nu_1\). We first show that \(\nu_1\) is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all \(0\leq\nu<\nu_1\), there exists an optimal measure \(\mu\geq0\) giving the lowest possible eigenvalue at fixed mass \(\mu(\mathbb{R}^3)=\nu\), which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.