Spectral correspondences for negatively curved Riemannian locally symmetric spaces

负弯曲黎曼局部对称空间的谱对应

• 批准号: 432944415
• 负责人: Professor Dr. Joachim Hilgert
• 金额: --
• 依托单位: Institut for Matematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/432944415?language=en
• 关键词: Spectral; correspondences; negatively; curved; Riemannian

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.

该项目的中心目标是使用这些共振和量子共振之间的（待建立的）对应关系来描述局部对称空间的波利科特-鲁埃尔共振。在经典力学和量子力学的描述中，负弯曲黎曼局部对称空间上的自由粒子的动力学性质之间存在密切的联系。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图来描述。对于紧致和凸协紧双曲曲面，这种对应图很好理解，并且对于给定的光谱参数，可以得出经典谐振态空间和量子谐振态空间之间的线性同构。对于更高维度的负弯曲局部对称空间，在紧致和协紧曲面上扩展结果存在许多障碍。这些障碍之一是泊松变换仅对于通用参数是可逆的，泊松变换取决于所谓的谱参数，并且在我们想要研究的谱对应现象的所有描述中至关重要。在表面的情况下，也可以为特殊参数建立光谱对应关系，因为两侧都有足够的明确信息以避免使用泊松变换。该项目的主要目标是将特殊谱参数的谱对应关系从紧致和协紧双曲曲面的情况扩展到一般的一阶局部对称空间。特别是寻找特殊光谱参数所携带的拓扑信息以及它们在描述 Selberg zeta 函数除数中所扮演的角色。人们可以希望获得关于在没有强对称条件的情况下可变负曲率流形的情况下可能成立的线索。