On the Schrödinger equation with potential in modulation spaces

Abstract

This work deals with Schrodinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations, following the footsteps of the preceding works (Cordero et al., Generalized metaplectic operators and the Schrodinger equation with a potential in the Sjostrand class, 2013; Cordero et al., Propagation of the Gabor wave front set for Schrodinger equations with non-smooth potentials, 2013). To the best of our knowledge these are the pioneering papers which contain the most general results about the time–frequency concentration of the Schrodinger evolution. 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.