Title: The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points,
Abstract: We study the adjoint of the double layer potential associated with the
Laplacian (the adjoint of the Neumann-Poincaré 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.