Intertwining the geodesic flow and the Schrodinger group on hyperbolic surfaces

We construct an explicit intertwining operator $\lcal$ between the Schrödinger group $e^{it \frac\Lap2} $ and the geodesic flow $g^t$ on certain Hilbert spaces of symbols on the cotangent bundle $T^* \X$ of a compact hyperbolic surface $\X = Γ\backslash \D$. Thus, the quantization Op(\lcal^{-1} a) satisfies an exact Egorov theorem. The construction of $\lcal$ is based on a complete set of Patterson-Sullivan distributions.

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