Higher rank quantum-classical correspondence

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 $G/K$.

arXiv comment23 pages, 6 figures

References in corpus (11)

• Upper bound on the density of Ruelle resonances for Anosov flows
• Power spectrum of the geodesic flow on hyperbolic manifolds
• Fuchsian groups of the second kind and representations carried by the limit set
• Sharp polynomial bounds on the number of Pollicott-Ruelle resonances
• Pollicott-Ruelle resonant states and Betti numbers
• Fractal Weyl law for the Ruelle spectrum of Anosov flows
• Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum
• Ruelle-Taylor resonances of Anosov actions
• Dynamics of geodesics, and Maass cusp forms
• Symbolic dynamics and transfer operators for Weyl chamber flows: a class of examples
• Eigenfunctions of transfer operators and automorphic forms for Hecke triangle groups of infinite covolume

Cited by in corpus (1)

• Temperedness of locally symmetric spaces: The product case