Spectral correspondences for negatively curved Riemannian locally symmetric spaces

这个项目的中心目标是利用这些共振和量子共振之间的对应关系来描述局部对称空间的波利科-鲁埃尔共振。负弯曲黎曼局部对称空间中自由粒子的动力学性质在经典力学和量子力学的描述中有着密切的联系。这种联系可以用所谓的经典共振态和量子系统之间的对应映射来描述。对于紧和凸紧双曲曲面，这种对应映射被很好地理解，并导致给定谱参数的经典和量子共振态空间之间的线性同构。对于高维的负弯曲局部对称空间，在紧曲面和紧曲面上推广结果存在许多障碍。其中一个障碍是泊松变换，它依赖于一个所谓的光谱参数，在我们想要研究的光谱对应现象的所有描述中都是至关重要的，它只对一般参数是可逆的。在曲面的情况下，由于两侧有足够的显式信息以避免使用泊松变换，因此也可以为特殊参数建立光谱对应关系。本课题的主要目的是将特殊谱参数的谱对应从紧和紧双曲曲面扩展到一般的1阶局部对称空间。特别是寻找异常谱参数所携带的拓扑信息，以及它们在描述Selberg zeta函数的除数中所起的作用。我们可以希望得到一些提示，说明在没有强对称条件的变负曲率流形的情况下，什么是正确的。

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.