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We study the possible expansion of the electromagnetic field scattered by astrictly convex metallic nanoparticle with dispersive material parametersplaced in a homogeneous medium in a low-frequency regime as a sum of modesoscillating at complex frequencies (diverging at infinity), known in thephysics literature as the quasi-normal modes expansion. We show that such anexpansion is valid in the static regime and that we can approximate theelectric field with a finite number of modes. We then use perturbative spectraltheory to show the existence, in a certain regime, of plasmonic resonances aspoles of the resolvent for Maxwell's equations with non-zero frequency. We showthat, in the time domain, the electric field can be written as a sum of modesoscillating at complex frequencies. We introduce renormalised quantities thatdo not diverge exponentially at infinity.

Modal approximation for strictly convex plasmonic resonators in the time domain: the Maxwell's equations