Guillarmou's Normal Operator for Magnetic and Thermostat Flows

TL;DR

This paper extends Guillarmou's normal operator to thermostat and magnetic flows on Anosov manifolds, demonstrating their elliptic pseudodifferential nature and applying this to establish a stability estimate for the magnetic X-ray transform.

Contribution

It generalizes Guillarmou's normal operator to more complex flows, showing their elliptic pseudodifferential properties and enabling stability estimates.

Findings

Normal operators are elliptic pseudodifferential operators of order -1.

Generalization to thermostat and magnetic flows.

Proved a stability estimate for the magnetic X-ray transform.

Abstract

Guillarmou's normal operator over a closed Anosov manifold is analogous to the classical normal operator of the geodesic X-ray transform over manifolds with boundary. In this paper, we generalize this normal operator, under some dynamical assumptions, to thermostat flows as well as to the case of the magnetic flows. In particular, we show that these generalized normal operators are elliptic pseudodifferential operators of order -1 in each case. As an application, we prove a stability estimate for the magnetic X-ray transform.