  We study propagation of phase space singularities for the initial value
Cauchy problem for a class of Schr\"odinger equations. The Hamiltonian is the
Weyl quantization of a quadratic form whose real part is non-negative. The
equations are studied in the framework of projective Gelfand--Shilov spaces and
their distribution duals. The corresponding notion of singularities is called
the Gelfand--Shilov wave front set and means the lack of exponential decay in
open cones in phase space. Our main result shows that the propagation is
determined by the singular space of the quadratic form, just as in the
framework of the Schwartz space, where the notion of singularity is the Gabor
wave front set.
