Abstract: 

We study pseudodifferential operators on a compact hyperbolic surface using "Zelditch" quantisation, which provides an exact correspondence between symbols and kernels of pseudodifferential operators on a hyperbolic surface.  Motivated by questions in quantum chaos, we study the trace of a certain time-varying family of operators, $(A_2)^*A_1(t)$ where $A_1(t)$ has a symbol which evolves under geodesic flow.  We find such traces are related to the theory of decay of correlations in dynamical systems and prove conditions under which this trace decays at $t$ approaches plus/minus infinity. This is joint work with Andrew Hassell.