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We study spectral properties of the Neumann-Poincar\'e operator on planardomains with corners with particular emphasis on existence of continuousspectrum and pure point spectrum. We show that the rate of resonance atcontinuous spectrum is different from that at eigenvalues, and then derive amethod to distinguish continuous spectrum from eigenvalues. We performcomputational experiments using the method to see whether continuous spectrumand pure point spectrum appear on domains with corners. For the computations weuse a modification of the Nystr\"om method which makes it possible to constructhigh-order convergent discretizations of the Neumann-Poincar\'e operator ondomains with corners. The results of experiments show that all three possiblespectra, absolutely continuous spectrum, singularly continuous spectrum, andpure point spectrum, may appear depending on domains. We also prove rigorouslytwo properties of spectrum which are suggested by numerical experiments:symmetry of spectrum (including continuous spectrum), and existence ofeigenvalues on rectangles of high aspect ratio.

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