Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces

Nalini Anantharaman, Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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

Original languageEnglish (US)
Pages (from-to)1103-1156
Number of pages54
JournalMathematische Annalen
Issue number4
StatePublished - Aug 2012

ASJC Scopus subject areas

  • General Mathematics


Dive into the research topics of 'Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces'. Together they form a unique fingerprint.

Cite this