Gabor Representations of evolution operators

We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic type. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schr\"odinger-type propagators \cite{fio3}, reveal to be an equally efficient tool for representing solutions to hyperbolic and parabolic-type differential equations with constant coefficients. In fact, the Gabor matrix representation of the corresponding propagator displays super-exponential decay away from the diagonal.