Propagation of Gabor Wave Front Set for Schr\"odinger Equations

We consider a general class of Scr\"odinger equations in R^d for which the problem of propagation of singularities is studied. For Schr\"odinger equations it is well known that, unlike hyperbolic equations, the classical wave front set of the solution is generally not transported according to the Hamiltonian flow. Here we show that this neat result is instead recaptured for a different notion of wave front set, i.e. the Gabor wave front set, defined in terms of cones in phase space, as opposite to cones in the frequency domain. Remarkably, similar results hold in presence of pseudo-differential perturbations with rough symbols, which may be not even differentiable.