AI 中文总结
研究在贝雷津 - 托普利兹量子化下，利用两两对易量子可观测量的联合本征截面定义等距嵌入，将量子可观测量实现为\(L^2(\Lambda_{a_0})\)上收敛到乘法算子的序列，探讨谱含义并给出相关应用。
AI 中文摘要
我们考虑在闭凯勒流形的贝雷津 - 托普利兹量子化背景下的一组两两对易的量子可观测量，并假设阿诺德 - 刘维尔定理适用于它们的主符号。我们利用这些可观测量的联合本征截面来定义量子空间到\(L^2(\Lambda_{a_0})\)的等距嵌入，其中\(\Lambda_{a_0}\)是一个固定的刘维尔环面。这些嵌入使得包括一些由不连续函数定义的广泛类别的量子可观测量能够实现为\(L^2(\Lambda_{a_0})\)上强收敛到乘法算子的算子序列。我们讨论了这种收敛的谱含义，并给出了对李代数表示的收缩和量子可观测量的谱投影对的应用。
英文摘要
We consider a collection of pairwise commuting quantum observables in the setting of Berezin--Toeplitz quantization of a closed Kähler manifold and assume that the Arnold--Liouville theorem applies to their principal symbols. We use joint eigensections of these observables to define isometric embeddings of the quantum spaces into $L^2(Λ_{a_0})$, where $Λ_{a_0}$ is a fixed Liouville torus. These embeddings allow a broad class of quantum observables, including some defined by discontinuous functions, to be realized as sequences of operators on $L^2(Λ_{a_0})$ that converge strongly to multiplication operators. We discuss the spectral implications of this convergence and give applications to contractions of Lie algebra representations and to pairs of spectral projections of quantum observables.
Comments63 pages
