Spectral properties of interface problems for Maxwell systems

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The colour of light emitted from a laser is determined by the frequencies of vibrations of atoms. Similarly, plucking a string of a guitar causes the string to vibrate and produce a sound. Changing the length of the string or the material from which it is made will change the sound it produces. This is due to the fact that these two properties of the string fix how fast it vibrates which in turn determines the sound. The frequencies of light transmitted through a material will depend on the electromagnetic properties of the material. The same principle is used for such diverse tasks as analysing the composition of drugs or the atmosphere of distant planets. Spectral theory is the branch of mathematics that investigates the frequencies of the vibrations (the spectrum) of a physical system and as such plays a role in many different areas, both in everyday situations and in scientific research. This project will consider the propagation of electromagnetic waves, such as light, in materials. We wish to determine the frequencies of light that the material allows to propagate. A particular focus will be on so-called surface plasmons which can be generated at the interface of two different materials. Surface plasmons have potential applications in many fields, including medical imaging and quantum or optical computing devices, where exploiting their properties could lead to significant improvements in the speed of data transfer. We will consider the physically relevant situation where energy is lost (dispersed) when the wave travels through the material. Mathematically, this leads to a so-called non-selfadjoint setting for the problem.Many problems for which spectral properties have been studied are so-called selfadjoint problems, often systems with an underlying conserved quantity such as energy. This has been driven in large part due to the importance of the theory of selfadjoint operators in quantum mechanics which provided much of the impetus for the development of spectral theory in the 20th century. On the other hand, there are many physical problems, such as the one we consider here, where the system under consideration loses or gains energy and therefore does not fall into the category above, for example, problems of analysing the transition from stability to turbulence in fluid flows and many other problems in hydrodynamics, magnetohydrodynamics, composite materials, lasers and nuclear scattering. These problems are described by non-selfadjoint operators which have very different spectral properties from selfadjoint operators. This makes their study more complicated but leads to a variety of new and sometimes unexpected consequences.