Spectral correspondences for negatively curved Riemannian locally symmetric spaces

这个项目的中心目标是利用局部对称空间的Pollicott-Ruelle共振与量子共振之间的对应关系来描述这些共振。在经典和量子力学的描述中，负弯曲黎曼局部对称空间上自由粒子的动力学性质之间有着密切的联系。这种联系可以用经典系统和量子系统的所谓共振态之间的对应映射来描述。对于紧致和凸余紧双曲曲面，这种对应映射被很好地理解，并且对于给定的谱参数，它导致经典共振态和量子共振态空间之间的线性同构。对于高维的负曲局部对称空间，要将结果推广到紧曲面和余紧曲面上，存在许多障碍。其中一个障碍是，泊松变换只对一般参数是可逆的，它依赖于所谓的谱参数，并且在我们想要研究的谱对应现象的所有描述中都是至关重要的。在曲面的情况下，也可以建立特殊参数的谱对应，因为人们在两侧都有足够的显式信息来避免使用泊松变换。这个项目的主要目的是将特殊谱参数的谱对应从紧的和上紧的双曲曲面的情况推广到一般的一阶局部对称空间。特别地，人们寻找由例外谱参数携带的拓扑信息以及它们在描述Selberg Zeta函数的除数时所起的作用。人们可以希望得到在没有强对称条件的可变负曲率流形的情况下什么可能是真的的提示。

The central goal of this project is to describe Pollicott-Ruelle resonances of locally symmetric spaces using a - to be established - correspondence between these resonances and quantum resonances.There are close connections between the dynamical properties of a free particle on a negatively curved Riemannian locally symmetric space in the descriptions of classical and quantum mechanics. Such a connection can be described in terms of a correspondence map between so-called resonant states of the classical and the quantum system. For compact and convex cocompact hyperbolic surfaces this correspondence map is well understood and leads to linear isomorphisms between spaces of classical and quantum resonant states for given spectral parameters. For negatively curved locally symmetric spaces of higher dimension there are a number of obstacles to the extension of the results on compact and cocompact surfaces. One of these obstacles is that the Poisson transformation, which depends on a so-called spectral parameter and is of crucial importance in all descriptions of the spectral correspondence phenomena we want to study, is invertible only for generic parameters. In the case of surfaces the spectral correspondence could be established also for the exceptional parameters since one had enough explicit information on both sides to avoid the use of the Poisson transform. The main objective of this project is to extend the spectral correspondence for exceptional spectral parameters from the case of compact and cocompact hyperbolic surfaces to general locally symmetric spaces of rank one. In particular one looks for the topological information carried by the exceptional spectral parameters and the role they play in the description of the divisor of the Selberg zeta function. One can hope to obtain hints for what might be true in the case of manifolds of variable negative curvature without the strong symmetry conditions.