On the complex symmetry of the Dirichlet-to-Neumann operator

It is well-known that the Stratton-Chu formalism allows a complete reconstruction of the interior and exterior electromagnetic fields (E,H) inside and outside a simply connected isotropic domain Omega with smooth boundary partOmega merely by knowledge of the tangential field components (E^t, H^t) or the equivalent magnetic and electric currents on the boundary partOmega in the absence of sources. A still stronger statement, resulting from the inherent duality of both Maxwellpsilas equations and the Stratton-Chu formalism, is that only one tangential field component E^t or H^t (in other words only one equivalent magnetic or electric current), is needed in order to describe the complete field configuration in the absence of source terms, except notably under resonance conditions. It follows that there exists a linear operator relationship between the equivalent magnetic and electric currents, called the Dirichlet-to-Neumann or Poincare-Steklov operator. In (de la Bourdonnaye, 1995), it is indicated how the Poincare-Steklov operator can be extracted from the Calderon projectors by means of a Schur complement method (see also (Knockaert et al., 2008) for the 2D case).