Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces

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