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

Abstract

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 $\mu \ge 0$ 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 $0\le \nu <{\nu }_1$, there exists an optimal measure $\mu \ge 0$ 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.

Final version to appear in Proc. London Math. Soc

References in corpus (4)

• On the eigenvalues of operators with gaps. Application to Dirac operators
• Self-adjointness of Dirac operators via Hardy-Dirac inequalities
• The Scott correction in Dirac-Fock theory
• Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue

Cited by in corpus (5)

• Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas
• The Scott correction in Dirac-Fock theory
• Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue
• Morse inequalities for ordered eigenvalues of generic self-adjoint families
• Which nuclear shape generates the strongest attraction on a relativistic electron? An open problem in relativistic quantum mechanics