The plasmonic eigenvalue problem

Abstract

A plasmon of a bounded domain $\Omega \subset {\mathbb{R}}^n$ is a non-trivial bounded harmonic function on ${\mathbb{R}}^n\setminus \partial \Omega$ which is continuous at $\partial \Omega$ and whose exterior and interior normal derivatives at $\partial \Omega$ have a constant ratio. We call this ratio a plasmonic eigenvalue of $\Omega$. Plasmons arise in the description of electromagnetic waves hitting a metallic particle $\Omega$. 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.

22 pages; replacement 8-March-14: minor corrections; to appear in Review in Mathematical Physics

References in corpus (4)

• Theory of surface plasmons and surface-plasmon polaritons
• Plasmonic atoms and plasmonic molecules
• Perturbation theory for plasmonic eigenvalues
• Surface plasmon resonances of an arbitrarily shaped nanoparticle: High frequency asymptotics via pseudo-differential operators

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