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The study of time harmonic electromagnetic waves at a flat interface of dispersive (i.e. frequency dependent) media leads to a non-self-adjoint Maxwell operator pencil. A classical application is to surface plasmon polaritons at the interface of a dielectric and a metal. We assume that the interface is flat and the media in the two half spaces are spatially homogenous and unbounded. The dependence of the permittivity on the spectral parameter (frequency) is generally nonlinear. The whole spectrum consists of eigenvalues and the essential spectrum, but the various standard types of essential spectra do not coincide in all cases. The functional setting is such that the operator domain is not a subset of the range which brings about a difficulty in defining the discrete spectrum. The spectral analysis is complete in one and two dimensions; in collaboration with Malcolm Brown (Cardiff), Michael Plum (Karlsruhe), and Ian Wood (Canterbury). In the case of cubically (Kerr) nonlinear materials monochromatic solutions do not exist. In the one dimensional case we find polychromatic solutions in the form of an infinite series of odd harmonics with the frequency of the leading order term being a complex eigenvalue of the linear problem. This work is in collaboration with Max Hanisch (Halle) and Runan He (Madrid).
Christian Brennecke (Uni Bonn)