The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points

We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincar\'e operator), as a map on the boundary surface $\Gamma$ of a domain in $\mathbb{R}^3$ with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across $\Gamma$. In particular, if the domain is understood as an inclusion with complex permittivity $\epsilon$, embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when $(\epsilon+1)/(\epsilon-1)$ belongs to the resolvent set in energy norm. We study surfaces $\Gamma$ that have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On $L^2(\Gamma)$, i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum.