On the well-posedness of magnetic Schrödinger equations with unbounded potentials

We consider magnetic Schrödinger equations with sublinear magnetic potentials and subquadratic electric potentials on Rᵈ, as well as generalizations thereof. We obtain new results on the global well-posedness of the Cauchy problem with initial data in magnetic modulation spaces MᵖA (Rᵈ). Our results are achieved by approximating the solution in phase space using the magnetic Hamiltonian flow. This method includes the potentials as part of the generalized Schrödinger operator instead of treating them as perturbations, and thereby allows us to deal with unbounded potentials. For A 0, the space MᵖA (Rᵈ) reduces to the usual modulation space Mᵖ (Rᵈ), for which relevant known results for the usual Schrödinger equation can be recovered.