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Article Dans Une Revue Proceedings of the London Mathematical Society Année : 2021

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

Résumé

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 Schrödinger 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.

Dates et versions

hal-02503462 , version 1 (10-03-2020)

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Maria J. Esteban, Mathieu Lewin, Eric Séré. Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue. Proceedings of the London Mathematical Society, 2021, Proceedings of the London Mathematical Society, 123 (4), pp.345-383. ⟨10.1112/plms.12396⟩. ⟨hal-02503462⟩
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