  We study the Cauchy problem for an evolution equation of Schr\"odinger type.
The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form
with a pseudodifferential perturbation of negative order from Shubin's class.
We prove that the propagator is a Fourier integral operator of Shubin type of
order zero. Using results for such operators and corresponding Lagrangian
distributions, we study the propagator and the solution, and derive phase space
estimates for them.
