Spectral properties of interface problems for Maxwell systems

激光器发出的光的颜色是由原子的振动频率决定的。类似地，拨动吉他的弦会使弦振动并产生声音。改变琴弦的长度或制作琴弦的材料会改变它所发出的声音。这是因为弦的这两个特性决定了它振动的速度，而这又决定了声音。透过材料的光的频率将取决于材料的电磁特性。同样的原理被用于分析药物成分或遥远行星的大气等各种任务。光谱理论是数学的分支，研究物理系统的振动频率（光谱），因此在许多不同的领域中发挥作用，无论是在日常生活中还是在科学研究中。这个项目将考虑电磁波，如光，在材料中的传播。我们希望确定材料允许传播的光的频率。一个特别的焦点将是所谓的表面等离子体激元，它可以在两种不同材料的界面处产生。表面等离子体激元在许多领域具有潜在的应用，包括医学成像和量子或光学计算设备，其中利用它们的特性可以显著提高数据传输的速度。我们将考虑当波穿过材料时能量损失（分散）的物理相关情况。在数学上，这导致了一个所谓的非自伴问题。许多研究光谱性质的问题都是所谓的自伴问题，通常是具有基本守恒量（如能量）的系统。这在很大程度上是由于自伴算符理论在量子力学中的重要性，它为世纪谱理论的发展提供了很大的动力。另一方面，还有许多物理问题，例如我们在这里考虑的问题，其中所考虑的系统损失或获得能量，因此不属于上述类别，例如，分析流体流动中从稳定到湍流的过渡的问题，以及流体力学，磁流体力学，复合材料，激光和核散射中的许多其他问题。这些问题描述的非自伴运营商有非常不同的频谱特性自伴运营商。这使得他们的研究更加复杂，但也导致了各种新的，有时是意想不到的后果。

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.