  We study propagation of the Gabor wave front set for a Schr\"odinger equation
with a Hamiltonian that is the Weyl quantization of a quadratic form with
non-negative real part. We point out that the singular space associated to the
quadratic form plays a crucial role for the understanding of this propagation.
We show that the Gabor singularities of the solution to the equation for
positive times are always contained in the singular space, and that they
propagate in this set along the flow of the Hamilton vector field associated to
the imaginary part of the quadratic form. As an application we obtain for the
heat equation a sufficient condition on the Gabor wave front set of the initial
datum tempered distribution that implies regularization to Schwartz regularity
for positive times.
