Spectral structure of the Neumann--Poincare operator on thin domains in two dimensions

We consider the spectral structure of the Neumann--Poincare operators defined on the boundaries of thin domains of rectangle shape in two dimensions. We prove that as the aspect ratio of the domains tends to $infty$, or equivalently, as the domains get thinner, the spectra of the Neumann--Poincare operators are densely distributed in the interval $[-1/2,1/2]$.