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We consider the Neumann--Poincar\'{e} operator on a planar domain enclosed bytwo touching circular boundaries. This domain, which is a crescent-shapeddomain or touching disks, has a cusp at the touching point of two circles. Weanalyze the operator via the Fourier transform on the boundary circles of thedomain. In particular, we define a Hilbert space on which the operator isbounded, self-adjoint. We then obtain the complete spectral resolution of theNeumann--Poincar\'{e} operator. On both the crescent-shaped domain and touchingdisks, the Neumann--Poincar\'{e} operator has only absolutely continuousspectrum on the closed interval $[-1/2,1/2]$. As an application, we analyze theplasmon resonance on the crescent-shaped domain and touching disks.

Spectral analysis of the Neumann--Poincar\'{e} operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance