  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.
