  Metallic nano-structures characterised by multiple geometric length scales
support low-frequency surface-plasmon modes, which enable strong light
localization and field enhancement. We suggest studying such configurations
using singular perturbation methods, and demonstrate the efficacy of this
approach by considering, in the quasi-static limit, a pair of nearly touching
metallic nano-spheres subjected to an incident electromagnetic wave polarized
with the electric field along the line of sphere centers. Rather than
attempting an exact analytical solution, we construct the pertinent
(longitudinal) eigen-modes by matching relatively simple asymptotic expansions
valid in overlapping spatial domains. We thereby arrive at an effective
boundary eigenvalue problem in a half-space representing the metal region in
the vicinity of the gap. Coupling with the gap field gives rise to a mixed-type
boundary condition with varying coefficients, whereas coupling with the
particle-scale field enters through an integral eigenvalue selection rule
involving the electrostatic capacitance of the configuration. By solving the
reduced problem we obtain accurate closed-form expressions for the resonance
values of the metal dielectric function. Furthermore, together with an
energy-like integral relation, the latter eigen-solutions yield also
closed-form approximations for the induced-dipole moment and gap-field
enhancement under resonance. We demonstrate agreement between the asymptotic
formulas and a semi-numerical computation. The analysis, underpinned by
asymptotic scaling arguments, elucidates how metal polarization together with
geometrical confinement enables a strong plasmon-frequency redshift and
amplified near-field at resonance.
