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arXiv · 1308.5425

Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields

Charles L. Epstein·Leslie Greengard·Michael O'Neil
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Abstract

The Debye source representation for solutions to the time harmonic Maxwell equations is extended to bounded domains with finitely many smooth boundary components. A strong uniqueness result is proved for this representation. Natural complex structures are identified on the vector spaces of time-harmonic Maxwell fields. It is shown that in terms of Debye source data, these complex structures are uniformized, that is, represented by a fixed linear map on a fixed vector space, independent of the frequency. This complex structure relates time-harmonic Maxwell fields to constant-k Beltrami fields, i.e. solutions of the equation curl(E) = kE. A family of self-adjoint boundary conditions are defined for the Beltrami operator. This leads to a proof of the existence of zero-flux, constant-k, force-free Beltrami fields for any bounded region in R^3, as well as a constructive method to find them. The family of self-adjoint boundary value problems defines a new spectral invariant for bounded domains in R^3.

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Charles L. Epstein, Leslie Greengard, Michael O'Neil. 2013-08-25. Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields. https://arxiv.org/abs/1308.5425

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