标题： 电磁波导的几何谱性质 显示英文标题

标题： Geometric spectral properties of electromagnetic waveguides

摘要： 考虑一个参考均匀且各向同性的电磁波导，其横截面是单连通的，并嵌入在理想导体中。在这种情况下，当波导是直的时，与具有常数扭转（可能为零）相关的自伴麦克斯韦算子的谱位于实线上，并且相对于零对称，并在原点附近表现出谱间隙。此外，谱是纯本质谱，并包含0，0是一个无限重的特征值。在这项工作中，我们提出了关于几何变形（特别是弯曲和扭转）对麦克斯韦算子谱的影响的新结果。更准确地说，一方面，我们提供了关于曲率和扭转渐近行为的充分条件，以确保参考波导的本质谱得到保留。我们的方法依赖于一种类似于Birman-Schwinger的原则，这可能具有独立的兴趣，并适用于其他上下文。另一方面，我们给出了充分条件（特别是涉及波导横截面几何形状的条件），以便几何变形在本质谱的间隙内产生离散谱（即有限重的孤立特征值）。此外，我们还给出了一些关于这些离散特征值定位的结果。涉及横截面的充分条件随后进行了分析和数值研究。最后，我们检查了其在横截面形状变形下的稳定性，特别关注矩形横截面波导的情况。 显示英文摘要

摘要： Consider a reference homogeneous and isotropic electromagnetic waveguide with a simply connected cross-section embedded in a perfect conductor. In this setting, when the waveguide is straight, the spectrum of the associated self-adjoint Maxwell operator with a constant twist (which may be zero) lies on the real line and is symmetric with respect to zero and exhibits a spectral gap around the origin. Moreover, the spectrum is purely essential, and contains 0 which is an eigenvalue of infinite multiplicity. In this work, we present new results on the effects of geometric deformations, specifically bending and twisting, on the spectrum of the Maxwell operator. More precisely, we provide, on the one hand, sufficient conditions on the asymptotic behavior of curvature and twist that ensure the preservation of the essential spectrum of the reference waveguide. Our approach relies on a Birman-Schwinger-type principle, which may be of independent interest and applicable in other contexts. On the other hand, we give sufficient conditions (involving in particular the geometrical shape of the cross-section of the waveguide) so that the geometrical deformation creates discrete spectrum (namely isolated eigenvalues of finite multiplicity) within the gap of the essential spectrum. In addition, we give some results on the localization of these discrete eigenvalues. The sufficient condition involving the cross-section is then studied both analytically and numerically. Finally, we examine its stability under shape deformations of the cross-section, focusing in particular on the case of a waveguide with a rectangular cross-section.