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.

从激光发出的光的颜色取决于原子的振动频率。同样，拔出一串吉他会导致弦振动并产生声音。更改字符串的长度或制成的材料将改变其产生的声音。这是由于字符串的这两个属性修复了它振动的速度，从而决定了声音。通过材料传输的光的频率将取决于材料的电磁特性。相同的原理用于分析药物组成或遥远行星大气等各种任务。光谱理论是数学的分支，它研究了物理系统的振动（光谱）的频率，因此在日常情况和科学研究中都在许多不同领域发挥作用。该项目将考虑材料中电磁波（例如光）的传播。我们希望确定材料允许传播的光频率。特定的焦点将放在可以在两种不同材料的界面上产生的所谓表面等离子体。表面等离子体在许多领域都有潜在的应用，包括医学成像，量子或光学计算设备，在这些领域中利用其性质可能会导致数据传输速度的显着提高。我们将考虑在波浪通过材料传播时能量损失（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频道设置。研究了光谱属性的许多问题是所谓的自我关节问题，通常具有基本保守数量（例如能量）的系统。这在很大程度上是由于自助会操作员在量子力学中的重要性，这为20世纪的光谱理论开发提供了大部分动力。另一方面，存在许多身体问题，例如我们在这里考虑的问题，其中所考虑的系统会损失或获得能量，因此不会属于上述类别，例如，分析从稳定性到流体流中的稳定性到湍流的问题以及流体动力学，磁动力学，组合材料，组合材料，激光，激光，核散射和核散射。这些问题是由非频道操作员描述的，这些操作员与自助接合运算符具有截然不同的光谱属性。这使他们的研究更加复杂，但会带来各种新的，有时甚至是意想不到的后果。