MAXWELL EQUATIONS FOR NONSMOOTH MEDIA - FRACTAL-SHAPED AND POINT-LIKE OBJECTS

Maxwell's equations are considered for the case of nonsmooth media, such as fractal-shaped and pointlike objects. In the first case the standard method of fitting normal and tangential field components breaks down. This problem is solved by showing that a proper time evolution for this situation can be obtained as a limiting case of a family of smooth ones. Thus a nonsmooth case can always be approximated by a smooth one with arbitrary precision. It is, in general, not possible to modify Maxwell's equations by a boundary condition in a single point. Here this problem is circumvented by using the recently developed theory of generalized point interactions [J. F. van Diejen and A. Tip, J. Math. Phys. 32, 630 (1991)]. This results in a Pontryagin space setting leading to a unitary scattering matrix in the Hilbert space associated with the field energy. The latter can be constructed in such a way that, asymptotically, for low frequencies, the scattering cross section for standard Mie scattering is recovered.