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
UR - http://www.scopus.com/inward/record.url?scp=84863327872&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84863327872&partnerID=8YFLogxK
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 -