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In this paper, we develop a new representation for outgoing solutions to thetime harmonic Maxwell equations in unbounded domains in $\bbR^3.$ Thisrepresentation leads to a Fredholm integral equation of the second kind forsolving the problem of scattering from a perfect conductor, which does notsuffer from spurious resonances or low frequency breakdown, although itrequires the inversion of the scalar surface Laplacian on the domain boundary.In the course of our analysis, we give a new proof of the existence ofnon-trivial families of time harmonic solutions with vanishing normalcomponents that arise when the boundary of the domain is not simply connected.We refer to these as $k$-Neumann fields, since they generalize, to non-zerowave numbers, the classical harmonic Neumann fields. The existence of$k$-harmonic fields was established earlier by Kress.

Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations