Wave packet analysis of Schrödinger equations in analytic function spaces

We consider a class of linear Schr\"odinger equations in $\rd$, with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which moves according to the Hamiltonian flow. As a consequence, we get an exponentially sparse representation of the Schr\"odinger propagator in phase space, with respect to Gabor wave packets. Similar results were first proved by D. Tataru in the smooth category.