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

TL;DR

This paper extends the Debye source representation for Maxwell solutions to bounded domains, introduces complex structures on Maxwell fields, and establishes a link to Beltrami fields, providing new spectral invariants and constructive methods.

Contribution

It introduces a novel extension of Debye sources, identifies natural complex structures on Maxwell fields, and connects these to Beltrami fields with new boundary conditions and spectral invariants.

Findings

Proved strong uniqueness for Debye source representation in bounded domains.

Identified complex structures on Maxwell fields that are frequency-independent.

Established existence and constructive methods for zero-flux Beltrami fields.

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.