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Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces

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We construct an explicit intertwining operator \({\mathcal L}\) between the Schrödinger group \({e^{it \frac\triangle2}}\) and the geodesic flow on certain Hilbert spaces of symbols on the cotangent bundle T*X Γ of a compact hyperbolic surface X Γ = Γ\D. We also define Γ-invariant eigendistributions of the geodesic flow \({PS_{j, k, \nu_j,-\nu_k}}\) (Patterson-Sullivan distributions) out of pairs of \({\triangle}\) -eigenfunctions, generalizing the diagonal case j = k treated in Anantharaman and Zelditch (Ann. Henri Poincaré 8(2):361–426, 2007). The operator \({\mathcal L}\) maps \({PS_{j, k, \nu_j,-\nu_k}}\) to the Wigner distribution \({W^{\Gamma}_{j,k}}\) studied in quantum chaos. We define Hilbert spaces \({\mathcal H_{PS}}\) (whose dual is spanned by {\({PS_{j, k, \nu_j,-\nu_k}}\)}), resp. \({\mathcal H_W}\) (whose dual is spanned by \({\{W^{\Gamma}_{j,k}\}}\)), and show that \({\mathcal L}\) is a unitary isomorphism from \({\mathcal H_{W} \to \mathcal H_{PS}.}\)

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Authors and Affiliations

  1. Laboratoire de Mathématique, Université d’Orsay Paris XI, 91405, Orsay Cedex, France

    Nalini Anantharaman

  2. Department of Mathematics, Northwestern University, Evanston, IL, 60208-2730, USA

    Steve Zelditch

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  1. Nalini Anantharaman
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  2. Steve Zelditch
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Correspondence to Steve Zelditch.

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Research partially supported by NSF grant DMS-0904252. N. Anantharaman wishes to acknowledge the support of Agence Nationale de la Recherche, under the grants ANR-09-JCJC-0099-01 and ANR-07-BLAN-0361.

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Anantharaman, N., Zelditch, S. Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces. Math. Ann. 353, 1103–1156 (2012). https://doi.org/10.1007/s00208-011-0708-6

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  • DOI: https://doi.org/10.1007/s00208-011-0708-6

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