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On the planar Neumann--Poincar\'e inverse spectral problem

by H. Ammari and M. J. Colbrook

(Report number 2026-40)

Abstract
Does the Neumann--Poincaré spectrum determine a planar domain up to an extended M\"obius 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 \(m\ge1\), 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 M\"obius equivalence classes illustrate the result. Direct proofs for complete polarization data show what spectral weights add to the eigenvalues.

Keywords: Neumann--Poincaré operator; inverse spectral problem; isospectral domains; shape reconstruction; Lipschitz stability; Grunsky coefficients; M\"obius invariance; generalised polarization tensors.

@Techreport{AC26_1195,
  author = {H. Ammari and M. J. Colbrook},
  title = {On the planar Neumann--Poincar\'e inverse spectral problem},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-40.pdf },
  year = {2026}
}

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