Study on Geometric Analysis

1.我们发现了一种新的量子遍历公式，并建立了量子遍历与经典遍历之间的显著联系。量子力学系统中的一个关键概念是空间平均和时间平均。我们采用的方法是伪微分算子和傅立叶积分算子的几何分析。我们可以建立一个关于离散周期磁薛定谔算子谱的结构的定理。特别是，我们发现了一种用群代数来解释能带结构产生的机制。图的第二上同调群--自同构群起着重要的作用。研究图上与薛定谔算子相关的谱结构与算子代数之间的关系仍然是很有意义的。

1. We have found a new formulation of quantum ergodicity, and established a remarkable relationship between quantum erogodicity and classical ergodicity. A key concepts is the space average and time average in quantum mechnical systems. The method we took up is geometric analysis of pseudo differential operators and Fourier integral operators.2. We could establish a theorem on the structure of the spectrum of discrete periodic magnetic Schroedinger operators. In particular, we found a mechanism in terms of group algebras why the band structure comes out. The second cohomology group of graph-automorphism group plays an important role. It is still interesting to investigate the relation between the spectral structure and operator algebras associated with Schroedinger operators on a graph.