Spectrum of the Neumann–Poincaré Operator for Ellipsoids and Tunability

We consider tunability of the eigenvalues of the Neumann–Poincaré operator on ellipsoids. We show in particular that for any number λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm \lambda}$$\end{document} with -1/2<λ<1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${-1/2 < {\rm \lambda} < 1/2}$$\end{document} there is an ellipsoid (a prolate spheroid or an oblate spheroid) on which λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm \lambda}$$\end{document} is an eigenvalue of the Neumann–Poincaré operator. As a byproduct, we find that there is a domain in three dimensions, actually an oblate spheroid, on which 0 is an eigenvalue of the Neumann–Poincaré operator.