The plasmonic eigenvalue problem

Abstract

A plasmon of a bounded domain Ω ⊂ ℝn is a non-trivial bounded harmonic function on ℝn\∂Ω which is continuous at ∂Ω and whose exterior and interior normal derivatives at ∂Ω have a constant ratio. We call this ratio a plasmonic eigenvalue of Ω. Plasmons arise in the description of electromagnetic waves hitting a metallic particle Ω. We investigate these eigenvalues and prove that they form a sequence of numbers converging to one. Also, we prove regularity of plasmons, derive a variational characterization, and prove a second-order perturbation formula. The problem can be reformulated in terms of Dirichlet–Neumann operators, and as a side result, we derive a formula for the shape derivative of these operators.