The Inverse Problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

Plamen Stefanov, Yang Yang

Published 2016-07-29Version 1

We consider the Dirichlet-to-Neumann map $\Lambda$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, a magnetic field $A$ and a potential $q$. We show that we can recover the jet of $g,A,q$ on the boundary form $\Lambda$ up to a gauge transformation in a stable way. We also show that the lens relation is the canonical relation of $\Lambda$ away from the diagonal and that the light ray transforms of $A$ and $q$ are recoverable from $\Lambda$ in a stable way. We present applications for recovery of $A$ and $q$ in a logarithmically stable way, and uniqueness with partial data.