具有一般电荷分布的Dirac-Coulomb算子 II. 最低特征值
Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue
AI总结：
本文研究一般电荷分布下Dirac-Coulomb算子的最低特征值，证明在临界质量以下存在集中于零测度紧集上的最优测度，并揭示临界质量与Hardy型不等式最佳常数的联系。
摘要：
考虑由非负有限测度μ生成的Coulomb势$-μ\ast|x|^{-1}$。众所周知，在固定质量$μ(\mathbb{R}^3)=ν$时，相应Schrödinger算子$-Δ/2-μ\ast|x|^{-1}$的最低特征值在μ正比于delta时达到最小。本文研究对于Dirac算子$-iα\cdot\nabla+β-μ\ast|x|^{-1}$同样成立的猜想。在该课题的前期工作中，我们证明了当μ没有质量大于或等于1的原子时，该算子是自伴的，并且其特征值由min-max公式给出。这里我们考虑临界质量$ν_1$，低于该质量时，对于所有满足$μ(\mathbb{R}^3)<ν_1$的$μ\geq0$，最低特征值不会下潜到下半连续谱中。我们首先证明$ν_1$与一个新的尺度不变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$0\leqν<ν_1$，存在一个最优测度$μ\geq0$，在固定质量$μ(\mathbb{R}^3)=ν$时给出最低可能特征值，该测度集中在一个Lebesgue测度为零的紧集上。最后一个性质是利用Dirac算子的一个新的唯一延拓原理证明的。存在性证明基于集中紧性原理。
英文摘要：
Consider the Coulomb potential $-μ\ast|x|^{-1}$ generated by a non-negative finite measure $μ$. It is well known that the lowest eigenvalue of the corresponding Schrödinger operator $-Δ/2-μ\ast|x|^{-1}$ is minimized, at fixed mass $μ(\mathbb{R}^3)=ν$, when $μ$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-iα\cdot\nabla+β-μ\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $μ$ 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 $ν_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $μ\geq0$ with $μ(\mathbb{R}^3)<ν_1$. We first show that $ν_1$ is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all $0\leqν<ν_1$, there exists an optimal measure $μ\geq0$ giving the lowest possible eigenvalue at fixed mass $μ(\mathbb{R}^3)=ν$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
