Application of Non-Self-Adjoint Operator Theory to the Singularity Expansion Method (SEM) and the Eigenmode Expansion Method (EEM) in Acoustic and Electromagnetic Problems.

Abstract

Abstract : Eigenfunction expansions for nonselfadjoint operators are important for scalar and electromagnetic wave scattering. Two methods: the Singularity Expansion Method (SEM), and the Eigenmode Expansion Method (EEM) which had been developed separately were studied. Criteria for their validity were established; moreover, the poles of the Green's function of the SEM are in 1-1 correspondence with the zeros of the eigenvalues of the EEM. A constructive numerical process for determining the poles of the Green's function was developed. Among several other results was the establishment of variational principles for the spectrum of compact nonselfadjoint operators. Another research area was the singularity behavior of eigenfunction expansions of various Green's functions in electromagnetic theory. The principal result shows that in an eigenfunction of a typical Green's function the point singularity of a Green's function is represented by a layer of surface singularity. This characteristic is analogous to the Gibbs' phenomenon where the representation of a discontinuous function by an orthogonal expansion creates the spikes which are absent in the original function. (Author)