On the Well-posedness of Magnetic Schrödinger Equations with Unbounded Potentials

Abstract

We consider magnetic Schrödinger 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 Schrödinger 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 Schrödinger equation can be recovered.