高阶与分数阶海森堡不确定性原理尖锐稳定性的傅里叶方法：一个新视角

中文摘要 AI 辅助

我们证明，在傅里叶侧，尖锐的二阶海森堡不确定性原理（HUP）及其对函数的尖锐稳定性，不过是应用于其傅里叶变换的一阶 $L^{2}$-Caffarelli--Kohn--Nirenberg 不等式及其尖锐稳定性。这给出了我们最近通过更长的论证建立的尖锐稳定性估计的几行证明。同样的观察随后产生了两个单参数族的尖锐分数阶海森堡不确定性原理，其中梯度被拉普拉斯算子的分数幂取代：对每个族，我们计算尖锐常数，刻画所有优化子，并建立尖锐稳定性估计。第一族的稳定性常数对于每个非负阶和每个维度都等于 $1$。负阶（此时分数拉普拉斯变为 Riesz 势）也被处理。最后，我们证明亏量携带的信息远多于到优化子的距离：它控制一个显式的连续余项链，其常数是一个显式算子的连续谱间隙。对于经典海森堡不确定性原理，这给出三个具有最优显式常数的余项，而对于二阶 HUP，则给出一个由四个余项组成的链，这些余项针对显式的合流超几何轮廓进行度量，并通过一组单一参数耦合。作为应用，我们还建立了无旋向量场的 HUP 稳定性链。

英文摘要

We show that, on the Fourier side, the sharp second order Heisenberg Uncertainty Principle (HUP) and its sharp stability for functions are nothing but the first order $L^{2}$-Caffarelli--Kohn--Nirenberg inequality and its sharp stability, applied to their Fourier transforms. This gives a proof in a few lines of the sharp stability estimate that we established recently by a much longer argument. The same observation then produces two one-parameter families of sharp fractional Heisenberg Uncertainty Principles, in which the gradient is replaced by a fractional power of the Laplacian: for each of them we compute the sharp constants, characterize all the optimizers, and establish the sharp stability estimates. The stability constant of the first family is equal to $1$ for every nonnegative order and every dimension. Negative orders, where the fractional Laplacian becomes a Riesz potential, are treated as well. Finally, we show that the deficit carries much more information than the distance to the optimizers alone: it controls an explicit chain of successive remainder terms whose constants are the successive spectral gaps of an explicit operator. For the classical Heisenberg Uncertainty Principle this gives three remainder terms with optimal explicit constants, and for the second order HUP a chain of four remainder terms measured against explicit confluent hypergeometric profiles and coupled through one single set of parameters. As applications, we also establish the chain of stability of the HUP for curl-free vector fields.

发表机构

• University of Connecticut（康涅狄格大学）
• Memorial University of Newfoundland（纽芬兰纪念大学）
• FPT University（FPT 大学）

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