Maxwell’s equations for nonsmooth media; fractal-shaped and pointlike objects

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Abstract
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. 3 2, 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.

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