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

We consider the spectral structure of the Neumann--Poincaré 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--Poincaré operators are densely distributed in the interval $[-1/2,1/2]$.