Higher rank quantum-classical correspondence

TL;DR

This paper establishes a quantum-classical correspondence for higher rank symmetric spaces, determining Ruelle-Taylor resonance locations, providing bounds, and proving a uniform spectral gap across different spaces.

Contribution

It introduces a novel quantum-classical correspondence linking Ruelle-Taylor resonances with eigenfunctions on higher rank symmetric spaces, and proves uniform spectral gaps.

Findings

Determined the location of Ruelle-Taylor resonances for higher rank spaces.

Provided a Weyl-lower bound on resonance counting functions.

Established a uniform spectral gap for irreducible higher rank spaces.

Abstract

For a compact Riemannian locally symmetric space $\Gamma\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $\Gamma$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$.