# Abstract

In the presence of absorption Maxwell's equations for a macroscopic
dielectric show a time convolution. In particular, the time evolution is
unidirectoral. We show how Maxwell's equations can be extended to a larger
set of equations with a unitary time evolution exp[iKt] in a suitable
Hilbert space. We then consider the situation where spatial periodicity is
present. Such systems are called "photonic crystals" and are at present
intensely studied in view of technological applications. After making a
Bloch-Floquet decomposition and a complex dilatation we find, using
analyticity and compactness argument, that the complex dilated version of K
has spectrum consisting of areas in the lower complex half plane. We shall
present a few numerical results for a two-dimensional case. Some physical
implications are also discussed.
Heinz Siedentop