量子化与过程
Quantization and process
- Université de Reims Champagne-Ardenne（兰斯香槟-阿登大学）
- Université Côte d’Azur（蔚蓝海岸大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究 $L^2(\mathbb{R}^n)$ 中伪微分算子的推广，基于相空间有界测度构建通用量子化演算，推导合成公式，证明特定 Lévy 过程测度满足相关假设，还得到 Weyl 演算的标准符号类及 Banach 代数同构等结果。
AI中文摘要：
本文研究 $L^2(\mathbb{R}^n)$（$n\geq1$）中伪微分算子的推广。新演算的定义仅依赖于相空间 $\mathbb{R}^{2n}$ 上的有界测度，每个测度对应一种特定的演算；反 Wick 量子化、Weyl 量子化、经典量子化及 Born-Jordan 量子化均为该通用演算的特例。随后研究该框架下的符号类：参数为1/2的 Gevrey 类是所有通用演算共有的符号类，即与参数化量子化的有界测度无关的符号类。若要考虑更大的符号类 $L^{\infty}(\mathbb{R}^{2n})$，需对测度施加精确的附加假设，该结果可应用于反 Wick 量子化，但不适用于 Weyl 量子化；对于 Weyl 伪微分演算，可得到标准的 Sjöstrand 类与 Gröchening 类。接着证明：相空间 $\mathbb{R}^{2n}$ 上扩散项大于1/4的 Lévy 过程的概率测度，是满足上述附加假设、可用于处理 $L^{\infty}(\mathbb{R}^{2n})$ 符号的自然测度实例，该关系通过 Lévy-Khintchine 公式推导得到。随后研究该通用框架下的合成律：为此，给出某精确符号类中两个符号的合成公式，适用于所有通用量子化。通用演算依赖于 Wick 量子化，因此针对某些精确符号类对其进行了初步考察，还提供了该语境下的附加结果，如 Mizrahi 级数展开及算子与符号类之间的 Banach 代数同构。
英文摘要：
This article is concerned with generalizations of pseudo-differential operators in $L^2(\mathbb{R}^n)$, $n\geq 1$. The definition of the new calculi depends only on bounded measures on the phase space $\mathbb{R}^{2n}$ and each measure gives rise to a specific calculus. Quantizations of anti-Wick, Weyl, classical and Born-Jordan are particular cases of the general calculi. Classes of symbols in this framework are then studied. The Gevrey class of parameter 1/2 is a class of symbol that is common to all the general calculi, that is, a class of symbols independent on the bounded measures parametrizing the quantizations. Precise additional hypotheses on the measures are necessary in the aim to consider the larger class of symbols $L^{\infty}(\mathbb{R}^{2n})$. This result can be applied for anti-Wick but not for Weyl quantization. Concerning Weyl pseudo-differential calculus, we recover the standard class of Sjöstrand and Gröchening. Then, we prove that probability measures of Lévy processes on the phase space $\mathbb{R}^{2n}$ with diffusion larger than 1/4 are natural examples of measures satisfying the latter additional hypotheses in order to consider $L^{\infty}(\mathbb{R}^{2n})$ symbols. This relation is derived using the Lévy-Khintchine formula. Composition laws in that general context are next investigated. In that purpose, we give a formula for the composition of two symbols in some precise class of symbols valid for all general quantizations. General calculi are relying on Wick quantization which is therefore primarily examined for some precise classes of symbols. Additional results in that context are provided, such as Mizrahi series expansions and Banach algebra isomorphisms between operators and symbol classes.
