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We analyze the spectrum of the Neumann-Poincaré (NP) operator for a doubly connected domain lying between two level curves defined by a conformal mapping, where the inner boundary of the domain is of general shape. The analysis relies on an infinite-matrix representation of the NP operator involving the Grunsky coefficients of the conformal mapping and an application of the Gershgorin circle theorem. As the thickness of the domain shrinks to zero, the spectrum of the doubly connected domain approaches the interval [−1/2, 1/2] in the Hausdorff distance and the density of eigenvalues approaches that of a thin circular annulus.
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Positivity conditions on the annulus via the double-layer potential kernel
We introduce and study a scale of operator classes on the annulus that is motivated by the Cρ classes of ρ-contractions of Nagy and Foiaş. In particular, our classes are defined in terms of the contractivity of the double-layer potential integral operator over the annulus. We prove that if, in addition, complete contractivity is assumed, then one obtains a complete characterization involving certain variants of the Cρ classes. Recent work of Crouzeix–Greenbaum and Schwenninger–de Vries allows us to also obtain relevant K-spectral estimates, generalizing and improving existing results from the literature on the annulus. Finally, we exhibit a special case where these estimates can be significantly strengthened.
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- Award ID(s):
- 2154494
- PAR ID:
- 10610982
- Publisher / Repository:
- Institute of Mathematics, Polish Academy of Sciences
- Date Published:
- Journal Name:
- Studia Mathematica
- Volume:
- 278
- Issue:
- 3
- ISSN:
- 0039-3223
- Page Range / eLocation ID:
- 233 to 265
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4064/sm231023-2-8
