Metallic nano-structures characterised by multiple geometric length scales support low-frequency
surface-plasmon modes, which enable strong light localisation and field enhancement. We suggest
to study 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 nanospheres
subjected to an incident electromagnetic wave polarised with the electric field along the line
of sphere centres. 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 induceddipole
moment and gap-field enhancement under resonance. We demonstrate agreement between the
asymptotic formulae and a semi-numerical computation. The analysis, underpinned by asymptotic
scaling arguments, elucidates how metal polarisation together with geometrical confinement enables
a strong plasmon-frequency redshift and amplified near-field at resonance.