Classification of spectra of the Neumann-Poincar\\'e operator on planar\n domains with corners by resonance

We study spectral properties of the Neumann-Poincar\\'e operator on planar\ndomains with corners with particular emphasis on existence of continuous\nspectrum and pure point spectrum. We show that the rate of resonance at\ncontinuous spectrum is different from that at eigenvalues, and then derive a\nmethod to distinguish continuous spectrum from eigenvalues. We perform\ncomputational experiments using the method to see whether continuous spectrum\nand pure point spectrum appear on domains with corners. For the computations we\nuse a modification of the Nystr\\"om method which makes it possible to construct\nhigh-order convergent discretizations of the Neumann-Poincar\\'e operator on\ndomains with corners. The results of experiments show that all three possible\nspectra, absolutely continuous spectrum, singularly continuous spectrum, and\npure point spectrum, may appear depending on domains. We also prove rigorously\ntwo properties of spectrum which are suggested by numerical experiments:\nsymmetry of spectrum (including continuous spectrum), and existence of\neigenvalues on rectangles of high aspect ratio.\n