Micro-local analysis applied to geometry and mathematical physics

本研究项目的第一个目标是将微观局域分析技术应用于几何和数学物理。我们计划研究的第一个问题是与自伴算子有关的特征向量的渐近性质：伴随分布及其与经典流的关系。第二个方面是研究与经典动力学有关的离散谱的统计性质。这个问题在几何学、物理学和量子化学中都有一些重要的动机，在量子化学中，半经典近似和高能方法被广泛使用。在数学上，关于可积和混沌运动的大特征值的普遍统计，有几个猜想是公开的，我们希望能有所贡献。

The first objective of this research project is to apply techniques of micro-local analysis in geometry and mathematical physics. The first problem we plan to investigate concerns the asymptotic properties of eigenvectors associated to self-adjoint operators: associated distributions and their relations with the classical flow.A second aspect is the study of statistical properties of the discrete spectrum in relation with the classical dynamics. This question has some important motivations in geometry, physics and in quantum chemistry where the semi-classical approximation and high-energy methods are intensively used. In mathematics several conjectures are open concerning the universal statistics of large eigenvalues for integrable and chaotic motions and we hope to contribute.