  We consider a class of linear Schr\"odinger equations in R^d with rough
Hamiltonian, namely with certain derivatives in the Sj\"ostrand class
$M^{\infty,1}$. We prove that the corresponding propagator is bounded on
modulation spaces. The present results improve several contributions recently
appeared in the literature and can be regarded as the evolution counterpart of
the fundamental result of Sj\"ostrand about the boundedness of
pseudodifferential operators with symbols in that class. Finally we consider
nonlinear perturbations of real-analytic type and we prove local wellposedness
of the corresponding initial value problem in certain modulation spaces.
