Schr\"odinger equations with rough Hamiltonians

Abstract

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.