General aspects of solving Helmholtz's equation underlying eigenvalue and scattering problems in electromagnetic wave theory

This paper considers conceptual aspects of different numerical methods for solving eigenvalue and scattering problems in electrodynamics. We will focu s on the separation of variables method, the method of lines as a special f inite-difference technique, and surface integral equation methods. It will be shown that there are interrelations between these different methods, and that it is possible to derive a general mathematical description based pn the separation of variables method. The method of lines is used as a starti ng point. Its concept is developed up to the Green's function belonging to the homogeneous Helmholtz equation with inhomogeneous boundary conditions p rescribed on surfaces which doesn't coincide with a constant coordinate lin e in one of the coordinate systems in which the Helmholtz equation becomes separable. With this Green's function we are able to derive a corresponding surface integral equation. Finally, the limiting behavior of the method of lines for an infinite number of discretization lines is considered. This r esults in a generalization of the separation of variables method.