Some integral geometry problems for wave equations

TL;DR

This paper investigates how solutions to wave equations on Minkowski spacetime can be uniquely determined from integrals along null geodesics, providing new proofs and stability results for inverse problems involving hyperbolic operators.

Contribution

It offers a new proof of stable determination for the Cauchy problem and extends stable determination results to sources with space-like singularities using microlocal analysis.

Findings

Stable determination of solutions from null geodesic integrals for Cauchy problems

Extension of stability results to sources with space-like singularities

Application of microlocal analysis to the normal operator of the light ray transform

Abstract

We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along null geodesics. For the Cauchy problem, we give a new proof of the stable determination result obtained in Vasy and Wang [12]. For the source problem, we obtain stable determination for sources with space-like singularities. Our proof is based on the microlocal analysis of the normal operator of the light ray transform composed with the parametrix for strictly hyperbolic operators.