Embedded eigenvalues for the Neumann-Poincar\'e operator
Spectral Theory
2019-03-05 v4 Classical Analysis and ODEs
Abstract
The Neumann-Poincar\'e operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincar\'e operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincar\'e operator for curves of class with the essential spectrum generated by a corner. Eigenvalues corresponding to even (odd) eigenfunctions are proved to lie within the essential spectrum of the odd (even) component of the operator when a curve is perturbed by inserting a small corner.
Cite
@article{arxiv.1806.00950,
title = {Embedded eigenvalues for the Neumann-Poincar\'e operator},
author = {Wei Li and Stephen P. Shipman},
journal= {arXiv preprint arXiv:1806.00950},
year = {2019}
}