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arXiv 2012.12667math-phmath.FAmath.MP
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夏普二阶不确定原理

Sharp second order uncertainty principles

Cristian Cazacu, Joshua Flynn, Nguyen Lam

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25 pages
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摘要：

研究了欧氏空间$\mathbb{R}^{N}$中的Caffarelli-Kohn-Nirenberg型二阶不等式，其中$N$表示维数.这种分析等价于对特殊类向量场的不确定原理的研究。特别是，我们表明，当从标量场$u: \rr^n\rightarrow \mathbb{C}$切换到矢量场的形式$\vec{u}:=\nabla U$（$U$是一个标量场）的海森堡不确定性原理（HUP）的最佳常数从$\frac{N^{2}}{4}$增加到$\frac{(N+2)^{2}}{4}$，和氢不确定性原理（HyUP）的最佳常数从$\frac{\left( N-1\right)^{2}}{4}$提高到$\frac{(N+1)^{2}}{4}$。由于我们的结果，我们回答了Maz'ya（积分方程算子理论2018）在$N=2$情况下关于无发散向量场的HUP的开放问题。

英文摘要：

We study sharp second order inequalities of Caffarelli-Kohn-Nirenberg type in the euclidian space $\mathbb{R}^{N}$, where $N$ denotes the dimension. This analysis is equivalent to the study of uncertainty principles for special classes of vector fields. In particular, we show that when switching from scalar fields $u: \rr^n\rightarrow \mathbb{C}$ to vector fields of the form $\vec{u}:=\nabla U$ ($U$ being a scalar field) the best constant in the Heisenberg Uncertainty Principle (HUP) increases from $\frac{N^{2}}{4}$ to $\frac{(N+2)^{2}}{4}$, and the optimal constant in the Hydrogen Uncertainty Principle (HyUP) improves from $\frac{\left( N-1\right)^{2}}{4}$ to $\frac{(N+1)^{2}}{4}$. As a consequence of our results we answer to the open question of Maz'ya (Integral Equations Operator Theory 2018) in the case $N=2$ regarding the HUP for divergence free vector fields.

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