Complete splitting of planar Robin eigenvalues by a single shrinking hole

Henry Zweiman · September 16, 2026 · P52 · Version 1.0

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For a prescribed point in a bounded connected planar Lipschitz domain and a fixed nonzero real Robin parameter, the paper constructs one analytic, strictly convex, area-preserving hole shape near a disk that completely splits every fixed eigenvalue cluster when the hole is sufficiently small. The same shape works across all finite spectral windows. The size threshold depends on the window. The proof includes arbitrary-hole coefficients at every nodal order, signed nonnodal separation and quantitative correctors.

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The spectral projection method, corrected Schur-complement mechanism, shape differentiation of polarization tensors and standard functional estimates are credited prior work. Some older full texts remain incompletely compared. No common positive size threshold for the entire infinite spectrum is proved. The mixed nonnodal Neumann problem is outside the theorem.

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