A complex-scaled boundary integral equation for the embedded eigenvalues and complex resonances of the Neumann-Poincaré operator on domains with corners

Abstract

The adjoint of the harmonic double-layer operator, also known as the Neumann-Poincaré (NP) operator, is a boundary integral operator whose real eigenvalues are associated with surface modes that find applications in e.g. photonics. On 2D domains with corners, the NP operator looses its compactness, as each corner induces a bounded interval of essential spectrum, and can exhibit both embedded eigenvalues and complex resonances. This work introduces a non-self-adjoint boundary integral operator whose discrete spectrum contains both embedded eigenvalues and complex resonances of the NP operator. This operator is obtained using a Green's function that is complex-scaled at each corner of the boundary. Numerical experiments using a Nyström discretization on a graded mesh demonstrates the accuracy of the method and its advantage over a 2D finite element discretization implementing the same complex scaling technique.