Geodesic flow and decay of traces on hyperbolic surfaces

We study pseudodifferential operators on a hyperbolic surface using `Zelditch quantization’. We motivate and study the trace of $A_2^{\ast }{A}_1(t)$, where $A_2$ is a fixed operator and the Zelditch symbol of $A_1(t)$ evolves by geodesic flow. We find conditions under which the trace decays exponentially as $t\to \pm \mathrm{\infty }$.