  This work deals with Schr\"odinger equations with quadratic and sub-quadratic
Hamiltonians perturbed by a potential. In particular we shall focus on bounded,
but not necessarily smooth perturbations. We shall give a representation of
such evolution as the composition of a metaplectic operator and a
pseudodifferential operator having symbol in certain classes of modulation
spaces. About propagation of singularities, we use a new notion of wave front
set, which allows the expression of optimal results of propagation in our
context. To support this claim, many comparisons with the existing literature
are performed in this work.
