On the planar Neumann--Poincaré inverse spectral problem
Abstract
Does the Neumann--Poincaré spectrum determine a planar domain up to an extended Möbius transformation? We construct a pair showing that the answer is no, even for analytic, strictly convex, algebraic boundaries. Their operators are similar. We then prove a positive result for prescribed finite-dimensional families of normalised exterior conformal maps. For every m≥1, the first m positive eigenvalues determine m real Laurent coefficients on a small truncated convex cone, provided the higher coefficients are a prescribed analytic remainder of quadratic order. Recovery is Lipschitz stable, uniformly down to the disk. The proof combines the Grunsky representation with a nondegeneracy argument for weighted Hankel matrices and differentiated perturbation estimates. Explicit two- and three-parameter examples, single-mode families with repeated leading eigenvalues, and Möbius equivalence classes illustrate the result. Direct proofs for complete polarization data show what spectral weights add to the eigenvalues.
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BibTeX
@misc{ammari2026planar,
title = {{On the planar Neumann--Poincaré inverse spectral problem}},
author = {Habib Ammari and Matthew J. Colbrook},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29661},
eprint = {2610.29661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29661},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the planar Neumann--Poincaré inverse spectral problem
AU - Habib Ammari
AU - Matthew J. Colbrook
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29661
UR - https://arxiv.org/abs/2610.29661
AB - Does the Neumann--Poincaré spectrum determine a planar domain up to an extended Möbius transformation? We construct a pair showing that the answer is no, even for analytic, strictly convex, algebraic boundaries. Their operators are similar. We then prove a positive result for prescribed finite-dimensional families of normalised exterior conformal maps. For every m≥1, the first m positive eigenvalues determine m real Laurent coefficients on a small truncated convex cone, provided the higher coefficients are a prescribed analytic remainder of quadratic order. Recovery is Lipschitz stable, uniformly down to the disk. The proof combines the Grunsky representation with a nondegeneracy argument for weighted Hankel matrices and differentiated perturbation estimates. Explicit two- and three-parameter examples, single-mode families with repeated leading eigenvalues, and Möbius equivalence classes illustrate the result. Direct proofs for complete polarization data show what spectral weights add to the eigenvalues.
ER -