  Metallic bowtie-shaped nanostructures are very interesting objects in optics,
due to their capability of localizing and enhancing electromagnetic fields in
the vicinity of their central neck. In this article, we investigate the
electrostatic plasmonic resonances of two-dimensional bowtie-shaped domains by
looking at the spectrum of their Poincar\'e variational operator. In
particular, we show that the latter only consists of essential spectrum and
fills the whole interval $[0,1]$. This behavior is very different from what
occurs in the counterpart situation of a bowtie domain with only
close-to-touching wings, a case where the essential spectrum of the Poincar\'e
variational operator is reduced to an interval strictly contained in $[0,1]$.
We provide an explanation for this difference by showing that the spectrum of
the Poincar\'e variational operator of bowtie-shaped domains with
close-to-touching wings has eigenvalues which densify and eventually fill the
remaining intervals as the distance between the two wings tends to zero.

Analysis of PDEs