Wave front set of solutions to Schr\"{o}dinger equations with time-dependent magnetic fields

TL;DR

This paper analyzes the wave front set of solutions to Schrödinger equations with time-dependent magnetic fields, revealing the absence of singularities in fundamental solutions under certain conditions.

Contribution

It introduces a method using wave packet transform to determine the wave front set for Schrödinger equations with time-dependent magnetic fields, including non-small magnetic fields.

Findings

Wave front set of solutions is characterized for time-dependent magnetic fields.

Fundamental solutions in spatially decaying magnetic fields have no singularities.

Method extends understanding of Schrödinger equations with complex magnetic interactions.

Abstract

In this paper, we determine the wave front set of solutions to the Schr\"{o}dinger equation with time-dependent magnetic fields. We considered time-dependent and `not so small' magnetic fields through the method using the wave packet transform established by K. Kato, M. Kobayashi and S. Ito. Furthermore, we checked that the fundamental solution of the Schr\"odinger equation in a spatially decaying magnetic field has no singularities as a consequence of our result.