Spectral properties of interface problems for Maxwell systems

麦克斯韦系统界面问题的谱特性

• 批准号: EP/W007037/1
• 负责人: Ian Geoffrey Wood
• 金额: $ 2.47万
• 依托单位: University of Kent
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW007037%2F1
• 关键词: Spectral; properties; interface; problems; Maxwell

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.

激光器发出的光的颜色由原子的振动频率决定。类似地，拨动吉他的琴弦会导致琴弦振动并发出声音。改变琴弦的长度或制成琴弦的材料会改变它产生的声音。这是因为琴弦的这两个属性决定了它振动的速度，进而决定了声音。通过材料传输的光的频率取决于材料的电磁特性。同样的原理也适用于分析药物成分或遥远行星大气层等各种任务。谱理论是数学的一个分支，研究物理系统的振动频率（频谱），因此在许多不同领域（无论是在日常情况还是在科学研究中）发挥着作用。该项目将考虑电磁波（例如光）在材料中的传播。我们希望确定材料允许传播的光的频率。特别关注的是所谓的表面等离子体，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都有潜在的应用，包括医学成像和量子或光学计算设备，利用它们的特性可以显着提高数据传输速度。我们将考虑波穿过材料时能量损失（分散）的物理相关情况。从数学上讲，这导致了该问题的所谓非自共轭设置。许多研究谱特性的问题都是所谓的自共轭问题，通常是具有潜在守恒量（例如能量）的系统。这在很大程度上是由于量子力学中自伴算子理论的重要性，它为 20 世纪谱理论的发展提供了很大的动力。另一方面，有许多物理问题，例如我们在这里考虑的问题，所考虑的系统会失去或获得能量，因此不属于上述类别，例如分析从稳定到湍流的转变的问题流体流动以及流体动力学、磁流体动力学、复合材料、激光和核散射中的许多其他问题。这些问题由非自共轭算子描述，它们具有与自共轭算子非常不同的谱特性。这使得他们的研究变得更加复杂，但会导致各种新的、有时是意想不到的后果。