Propagation of singularities for semilinear Schr\"odinger equations

We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow, but for Sobolev regularities in a certain range and provided the notion of Sobolev-wave front set is conveniently modified. The proof makes use of a weighted version of the paradifferential calculus, adapted to our situation. The results can be regarded as the Schrodinger counterpart of those known for semilinear hyperbolic equations, which hold with the usual wave front set.