Other Solution Methods

Abstract

In Chap. 3 and 4 we became familiar with two different ways of approximating the Green functions related to the scattering problems by finite series expansions in terms of appropriate eigensolutions of the Helmholtz and vector-wave equation. Both of the described ways produce the same expressions. The thus approximated Green functions result in corresponding series expansions of the scattered field with expansion coefficients calculated via the T-matrix from the known expansion coefficients of the primary incident field at the scatterer surface. Demonstrating that the T-matrix is a decisive element of the relevant Green function and that some important properties of the T-matrix like symmetry and unitarity are related to corresponding properties of the Green function can be considered to be the most important results of these two chapters. But there still exist other solution methods for the scattering problem of our interest which have been derived historically from other principles and assumptions used in the foregoing two chapters. Surprisingly, this holds for the T-matrix method itself. It was originally developed by use of the so-called “Extended Boundary Condition.” In this chapter we will therefore answer the question of how such methods fit into the developed Green function formalism, or, alternatively, how we have to change the formalism appropriately to end up with some other solution methods. Thereby, it is not our intention to provide a description of selected solution methods which is as complete as possible. In fact, we are more interested in demonstrating that some of the solution methods developed originally from other principles and assumptions can be mapped onto the Green function formalism, and that this formalism provides therefore a sound mathematical basis to analyze the advantages and disadvantages as well as the capabilities of different solution methods. The following considerations are mainly restricted to the scalar case. An exception from this is made when dealing with the so-called “Lippmann–Schwinger” equations which will be derived for the scalar as well as the dyadic Green functions and interaction operators at the end of this chapter.