Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces

Abstract

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 $${\mathcal H_{W} \to \mathcal H_{PS}.}$$