On the Well-posedness of Magnetic Schr\"odinger Equations with Unbounded Potentials

Abstract

We consider magnetic Schr\"odinger equations with sublinear magnetic potentials and subquadratic electric potentials on $\mathbb{R}^{d}$, 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 $M^{p}_{A}(\mathbb{R}^{d})$. 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 Schr\"odinger operator instead of treating them as perturbations, and thereby allows us to deal with unbounded potentials. For $A \equiv 0$, the space $M^{p}_{A}(\mathbb{R}^{d})$ reduces to the usual modulation space $M^{p}(\mathbb{R}^{d})$, for which relevant known results for the usual Schr\"odinger equation can be recovered.