Cloaking by anomalous localized resonance via spectral analysis of a Neumann-Poincaré operator

The classical notion of Neumann-Poincar´e (NP) operator appears naturally when we attempt to solve Dirichlet or Neumann boundary value problems using layer potentials. In this talk, we will review some properties of NP-operators and show that they can be used to give a mathematical analysis of cloaking by anomalous localized resonance (CALR). More precisely, we shall discuss the following problem. If a body of dielectric material is coated by a plasmonic structure of negative dielectric constant with nonzero loss parameter, then CALR may occur as the loss parameter tends to zero. Anomalous localized resonance is the phenomenon of field blow-up in a localized region and it may (and may not) occur depending upon the structure and the location of the source. We will show that the eigenvalue distribution of the NP-operator associated with the structure plays a special role in the occurrence of CALR and we will give a necessary and sufficient condition on the source term for electromagnetic power dissipation to blow up as the loss parameter of the plasmonic material goes to zero. This condition is written in terms of the Newtonian potential of the source term and eigenfunctions of the associated NP operator.