Intertwining the geodesic flow and the Schrodinger group on hyperbolic surfaces

TL;DR

This paper constructs an explicit operator linking quantum evolution and classical geodesic flow on hyperbolic surfaces, enabling exact Egorov theorem via Patterson-Sullivan distributions.

Contribution

It introduces a novel explicit intertwining operator between Schrödinger evolution and geodesic flow on hyperbolic surfaces, based on Patterson-Sullivan distributions.

Findings

Established an exact Egorov theorem for the constructed operator.

Provided a new explicit intertwining operator for quantum-classical correspondence.

Connected Patterson-Sullivan distributions to quantum dynamics on hyperbolic surfaces.

Abstract

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.