Infinitely many embedded eigenvalues for the Neumann-Poincar\'e operator in 3D

TL;DR

This paper demonstrates the existence of infinitely many eigenvalues embedded within the essential spectrum of the Neumann-Poincaré operator on a specially constructed 3D surface with a conical singularity.

Contribution

It constructs a specific 3D surface with a conical singularity that causes the NP operator to have infinitely many embedded eigenvalues, advancing understanding of spectral properties.

Findings

Infinitely many eigenvalues are embedded in the essential spectrum.

Construction of a surface with a conical singularity induces this spectral property.

The approach relies on symmetry and small perturbations.

Abstract

This article constructs a surface whose Neumann-Poincar\'e (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.