INFINITELY MANY EMBEDDED EIGENVALUES FOR THE NEUMANN-POINCARE OPERATOR IN 3D

Authors

Wei Li, Depaul Univ, Dept Math Sci, Chicago, IL 60614 USAFollow
Karl-Mikael Perfekt, Norwegian Univ Sci & Technol NTNU, Dept Math Sci, NO-7491 Trondheim, NorwayFollow
Stephen P. Shipman, Louisiana State Univ, Dept Math, Baton Rouge, LA 70803 USAFollow

Author ORCID Identifier

Li, Wei https://orcid.org/0000-0003-4098-4705 Perfekt, Karl-Mikael https://orcid.org/0000-0003-4574-9683

Document Type

Article

Publication Date

2022

Abstract

This article constructs a surface whose Neumann-Poincare (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts the essential spectrum. Rotational symmetry allows a decomposition of the operator into Fourier components. Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator. The proof requires the perturbation to be sufficiently small, with controlled curvature, and the conical singularity to be sufficiently flat.

Publication Source (Journal or Book title)

SIAM JOURNAL ON MATHEMATICAL ANALYSIS

First Page

343

Last Page

362

Recommended Citation

Li, W., Perfekt, K., & Shipman, S. P. (2022). INFINITELY MANY EMBEDDED EIGENVALUES FOR THE NEUMANN-POINCARE OPERATOR IN 3D. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 54 (1), 343-362. https://doi.org/10.1137/21M1400365