  Localized surface plasmons are charge density oscillations confined to
metallic nanoparticles. Excitation of localized surface plasmons by an
electromagnetic field at an incident wavelength where resonance occurs results
in a strong light scattering and an enhancement of the local electromagnetic
fields. This paper is devoted to the mathematical modeling of plasmonic
nanoparticles. Its aim is threefold: (i) to mathematically define the notion of
plasmonic resonance and to analyze the shift and broadening of the plasmon
resonance with changes in size and shape of the nanoparticles; (ii) to study
the scattering and absorption enhancements by plasmon resonant nanoparticles
and express them in terms of the polarization tensor of the nanoparticle.
Optimal bounds on the enhancement factors are also derived; (iii) to show, by
analyzing the imaginary part of the Green function, that one can achieve
super-resolution and super-focusing using plasmonic nanoparticles. For
simplicity, the Helmholtz equation is used to model electromagnetic wave
propagation.
