Quantum-Classical Correspondences for Locally Symmetric Spaces

TL;DR

This paper reviews the current understanding of the relationships between spectral invariants of certain flows on locally symmetric spaces and their quantum counterparts, proposing future research directions.

Contribution

It summarizes the status of a program linking spectral invariants and quantizations on locally symmetric spaces and outlines concrete problems for advancing this research.

Findings

Reviewed existing correspondences between classical and quantum spectral invariants.

Identified key challenges and open problems in the study of these correspondences.

Proposed specific future research steps to complete the program.

Abstract

We review the status of a program, outlined and motivated in the introduction, for the study of correspondences between spectral invariants of partially hyperbolic flows on locally symmetric spaces and their quantizations. Further we formulate a number of concrete problems which may be viewed as possible further steps to be taken in order to complete the program.