Spectral correspondences for negatively curved Riemannian locally symmetric spaces

• Applicant: Professor Dr. Joachim Hilgert
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 432944415

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.

• DFG Programme: Research Grants
• International Connection: France, Luxembourg
• Co-Investigators: Professor Dr. Jan Frahm; Professor Dr. Tobias Weich
• Cooperation Partners: Professor Dr. Colin Guillarmou; Professor Dr. Martin Olbrich; Professorin Dr. Angela Pasquale