Consider the Coulomb potential $-\mu *{|x|}^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schrödinger operator $-\mathrm{\Delta }/2-\mu *{|x|}^{-1}$ 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 $-i\mathit{\alpha }·\nabla +\beta -\mu *{|x|}^{-1}$. 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 continuous spectrum for any $\mu ⩾0$ with $\mu ({\mathbb{R}}^3)<{\nu }_1$. We first show that ${\nu }_1$ is related to the best constant in a new scale-invariant Hardy-type inequality. Our main result is that for all $0⩽\nu <{\nu }_1$, there exists an optimal measure $\mu ⩾0$ 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.

中文翻译：

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具有一般电荷分布的 Dirac-Coulomb 算子 II。最低特征值

考虑库仑势 $-\mu *{|X|}^{-1}$ 由非负有限测度生成 $\mu$. 众所周知，对应的薛定谔算子的最低特征值$-\mathrm{\Delta }/2-\mu *{|X|}^{-1}$ 在固定质量下最小化 $\mu ({\mathbb{电阻}}^3)=\nu$， 什么时候 $\mu$与 delta 成正比。在本文中，我们研究了同样适用于狄拉克算子的猜想$-\mathrm{一世}\mathit{\alpha }·\nabla +\beta -\mu *{|X|}^{-1}$. 在之前关于这个主题的工作中，我们证明了这个算子是自伴随的，当$\mu$没有质量大于或等于 1 的原子，并且其特征值由 min-max 公式给出。这里我们考虑临界质量${\nu }_1$，低于该值的最低特征值不会深入到任何较低的连续谱中 $\mu ⩾0$ 和 $\mu ({\mathbb{电阻}}^3)<{\nu }_1$. 我们首先证明${\nu }_1$与新的尺度不变的 Hardy 型不等式中的最佳常数有关。我们的主要结果是，对于所有人$0⩽\nu <{\nu }_1$，存在最优测度 $\mu ⩾0$ 在固定质量下给出尽可能低的特征值 $\mu ({\mathbb{电阻}}^3)=\nu$，它集中在一个紧凑的 Lebesgue 测度零集上。最后一个属性是使用狄拉克算子的新的独特延续原则显示的。存在性证明是基于集中紧性原理。