Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold
Abstract
We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set S, and the resulting waves are observed on an open receiver set R, which may be at a positive distance from S. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: |ϕ|R|L2(R) ≥ ce−Cλθ|ϕ|L2(M) for every eigenfunction ϕ, where λ is the corresponding eigenvalue, θ∈[0,1/2), and the constants c,C>0 do not depend on ϕ. If a bound of this type holds also on S, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with θ=1/2. Uniqueness nevertheless fails at this endpoint: in every dimension n≥ 2, we construct a pair of nonisometric connected, closed Riemannian n-manifolds that produce identical wave observations.
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BibTeX
@misc{ylinen2026uniqueness,
title = {{Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold}},
author = {Lauri Ylinen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00871},
eprint = {2610.00871},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00871},
}
RIS
TY - UNPB
TI - Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold
AU - Lauri Ylinen
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00871
UR - https://arxiv.org/abs/2610.00871
AB - We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set S, and the resulting waves are observed on an open receiver set R, which may be at a positive distance from S. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: |ϕ|_R|_L^2(R) ≥ ce^−Cλ^θ|ϕ|_L^2(M) for every eigenfunction ϕ, where λ is the corresponding eigenvalue, θ∈[0,1/2), and the constants c,C>0 do not depend on ϕ. If a bound of this type holds also on S, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with θ=1/2. Uniqueness nevertheless fails at this endpoint: in every dimension n≥ 2, we construct a pair of nonisometric connected, closed Riemannian n-manifolds that produce identical wave observations.
ER -