On the planar Neumann--Poincaré inverse spectral problem

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 $m\ge 1$, 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.