论文标题
亚拉普拉斯通过关节光谱的量子限制
Quantum limits of sub-Laplacians via joint spectral calculus
论文作者
论文摘要
我们建立了有关某些亚拉普拉斯人的量子限制（QL）的两个结果。首先，在涉及次级laplacian定义的矢量场上的换算性假设下，我们证明可以将任何QL分为几个可以单独研究的部分，并且来自特征函数相关序列的良好特征部分。其次，在此结果的基础上，我们详细研究了特定的亚蓝石家族的QLS，该家族定义在海森堡群体的紧凑型商品中。我们通过测量结果的分解来表达QLS，这是从谐波振荡器出现的亚拉普拉斯元素的自然光谱分解之后表达的。这两个结果均基于与亚拉普拉斯元素通勤的适当椭圆算子以及相关的关节光谱演算的构建。他们说明了一个事实，即由于光谱中可能具有高变化性，亚拉普拉奇人的光谱理论非常丰富。
We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub- Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we study in detail the QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear. Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possible high degeneracies in the spectrum, the spectral theory of sub-Laplacians is very rich.
