论文标题
狄拉克 - 库仑运算符，具有一般性电荷分布。 ii。最低特征值
Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue
论文作者
论文摘要
考虑由非负有限度量$μ$生成的库仑电势$-μ\ ast | x |^{ - 1} $。众所周知，相应的schrödinger操作员的最低特征值$-Δ/2-μ\ ast | x |^{ - 1} $被最小化，以固定的质量$μ（\ mathbb {r}^3）=ν$，当$ $与Delta成比例时。在本文中，我们调查了以下猜想：dirac运算符$-Iα\ cdot \ nabla+β-β-μ\ ast | x |^{ - 1} $相同。在先前的有关该主题的工作中，我们证明了当$μ$的质量没有大于或等于1的原子时，该操作员是自动化的，并且其特征值由Min-Max公式给出。在这里，我们考虑了临界质量$ν_1$，在此下面，最低特征值不会潜入所有$μ\ geq0 $的较低连续体频谱，均以$μ（\ mathbb {r}^3）<ν_1$。我们首先表明$ν_1$与新的缩放不变性不平等不平等中的最佳常数有关。我们的主要结果是，对于所有$ 0 \leqν<ν_1$，都存在最佳度量$μ\ geq0 $，以固定质量$μ（\ Mathbb {r}^3）=ν$提供最低可能的特征值，该值集中在紧凑的Lebesgue Matues Matues Matues Matues Mesumy Zero上。最后一个属性使用新的独特延续原理为狄拉克操作员显示。存在证明是基于浓度 - 紧密度原理。
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.
