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

TL;DR

This paper establishes the global well-posedness of magnetic Schr"odinger equations with unbounded potentials by using phase space approximation via magnetic Hamiltonian flow, extending known results to more general potentials.

Contribution

It introduces a novel phase space approximation method that incorporates unbounded potentials directly into the Schr"odinger operator, enabling new well-posedness results.

Findings

Proves global well-posedness for magnetic Schr"odinger equations with unbounded potentials.

Extends modulation space techniques to magnetic Schr"odinger equations.

Recovers known results for the case without magnetic 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.