Spectral structure of the Neumann--Poincaré operator on tori
arXiv:1810.09693 · doi:10.1016/j.anihpc.2019.05.002
Abstract
We address the question whether there is a three-dimensional bounded domain such that the Neumann--Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann--Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.
14 pages
References in corpus (3)
- The essential spectrum of the Neumann--Poincare operator on a domain with corners
- Weyl's law for the eigenvalues of the Neumann--Poincaré operators in three dimensions: Willmore energy and surface geometry
- A concavity condition for existence of a negative Neumann-Poincaré eigenvalue in three dimensions