  We construct an explicit intertwining operator $\lcal$ between the
Schr\"odinger 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 $\X = \Gamma \backslash \D$. 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.
