Singularity Expansion Method and Complex Singularities of Exterior Scalar and Vector Scattering in Acoustics and Electromagnetic Theory.

摘要

An examination of the relationship between the scattering matrix (the Fourier transform of the scattering operator) and the integral equations used in the Singularity Expansion Method (SEM) established that only the complex poles off the axis are intrinsically associated with the scatterer. While it has been known in special cases that those on the axis do not contribute to the field, this appears to be the first time this relationship has been clearly exhibited. Since the scattering matrix can be shown to be analytic in a half-plane containing the axis, any integral equation should exhibit the same properties for this region. In particular, the integral equations of SEM have at most two eigenvalues + or - and these are functions of the at-most-denumerable number of complex eigenvalues of the associated differential equations. The theory of non-self-adjoint operators in Hilbert space could have significant implications for that part of the electrical engineering formalism known as the Eigenmode Expansion Method (EEM) particularly in relevance to the use of this formalism in developing equivalent circuits. (jhd)