Infinitely many embedded eigenvalues for the NeumannPoincaré operator in 3D
Abstract
This article constructs a surface whose NeumannPoincaré (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 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:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.04371
 Bibcode:
 2020arXiv200904371L
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Spectral Theory;
 31B10;
 45A05;
 45C05;
 45E05;
 45P05