TY - JOUR

T1 - Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces

AU - Anantharaman, Nalini

AU - Zelditch, Steve

N1 - Funding Information:
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.

PY - 2012/8

Y1 - 2012/8

N2 - We construct an explicit intertwining operator L between the Schrödinger group 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, vj,-vk (Patterson-Sullivan distributions) out of pairs of Δ-eigenfunctions, generalizing the diagonal case j = k treated in Anantharaman and Zelditch (Ann. Henri Poincaré 8(2):361-426, 2007). The operator L maps PS j, k, vj,-vk to the Wigner distribution W j, k Γ studied in quantum chaos. We define Hilbert spaces H PS (whose dual is spanned by {PS j, k, vj,-vk}), resp. H W (whose dual is spanned by {W j, k Γ}), and show that L is a unitary isomorphism from H W → H PS

AB - We construct an explicit intertwining operator L between the Schrödinger group 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, vj,-vk (Patterson-Sullivan distributions) out of pairs of Δ-eigenfunctions, generalizing the diagonal case j = k treated in Anantharaman and Zelditch (Ann. Henri Poincaré 8(2):361-426, 2007). The operator L maps PS j, k, vj,-vk to the Wigner distribution W j, k Γ studied in quantum chaos. We define Hilbert spaces H PS (whose dual is spanned by {PS j, k, vj,-vk}), resp. H W (whose dual is spanned by {W j, k Γ}), and show that L is a unitary isomorphism from H W → H PS

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U2 - 10.1007/s00208-011-0708-6

DO - 10.1007/s00208-011-0708-6

M3 - Article

AN - SCOPUS:84863327872

VL - 353

SP - 1103

EP - 1156

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 4

ER -