Magnetic Fourier Integral Operators

TL;DR

This paper extends Fourier Integral Operator theory to include magnetic fields, establishing composition, continuity, and Egorov theorems, and representing magnetic evolution groups as magnetic Fourier Integral Operators.

Contribution

It introduces a magnetic Fourier Integral Operator framework, generalizing classical theory to magnetic contexts and applying it to evolution groups of magnetic pseudodifferential operators.

Findings

Representation of magnetic evolution groups as magnetic Fourier Integral Operators

Proven composition and continuity theorems in magnetic Sobolev spaces

Derived estimates for the distribution kernel and propagation of singularities

Abstract

In some previous papers we have defined and studied a 'magnetic' pseudodifferential calculus as a gauge covariant generalization of the Weyl calculus when a magnetic field is present. In this paper we extend the standard Fourier Integral Operators Theory to the case with a magnetic field, proving composition theorems, continuity theorems in 'magnetic' Sobolev spaces and Egorov type theorems. The main application is the representation of the evolution group generated by a 1-st order 'magnetic' pseudodifferential operator (in particular the relativistic Schr\"{o}dinger operator with magnetic field) as such a 'magnetic' Fourier Integral Operator. As a consequence of this representation we obtain some estimations for the distribution kernel of this evolution group and a result on the propagation of singularities.