A Hyperbolic Inverse Problem for lower order terms on a closed manifold with disjoint data
Matti Lassas, Boya Liu, Teemu Saksala, Andrew Shedlock, Ziyao Zhao
TL;DR
The paper addresses the inverse problem of recovering time-independent lower order terms in the magnetic Schrödinger perturbation of the wave equation on a known closed Riemannian manifold from local measurements in disjoint source and receiver regions. It employs the Boundary Control method, combined with Tataru's unique continuation and Blagoveščenskij's identities, to prove global uniqueness of the lower order terms up to a gauge, under the geometric control condition on the receiver set and a local symmetry condition for distances. The main result shows that if two coefficient pairs $(A_1,q_1)$ and $(A_2,q_2)$ yield the same local source-to-solution map at time $2T$, then $q_1=q_2$ and $A_1=A_2+i κ^{-1} dκ$ for a unitary $κ$ with $κ=1$ on $\mathcal{S}\cup\mathcal{R}$. The proof constructs point-concentrating waves via controllable sets, derives a global gauge function by patching local gauge data, and establishes the gauge-invariant recovery of the coefficients, extending previous overlapping-data results to the disjoint-data setting on closed manifolds.
Abstract
We study the unique recovery of time-independent lower order terms appearing in the symmetric first order perturbation of the Riemannian wave equation by sending and measuring waves in disjoint open sets of \textit{a priori} known closed Riemannian manifold. In particular, we show that if the set where we capture the waves satisfies a geometric control condition as well as a certain local symmetry condition for the distance functions, then the aforementioned measurement is sufficient to recover the lower order terms up to the natural gauge. For instance, our result holds if the complement of the receiver set is contained in a simple Riemannian manifold.