Robin波导中曲率诱导束缚态及其渐近性质
Curvature-induced bound states in Robin waveguides and their asymptotical properties
摘要：
我们分析了光滑边界的无限平面区域中Robin Laplacian的束缚态，特别是它们与边界几何的关系。考虑的域有局部直边界，例如，局部变形的半平面或楔形，或无限条，或者它们是有界障碍物的外部。在Robin条件是强吸引的情形下，我们得到了一个两项渐近公式，其中次领先项由边界曲率的极值决定。我们还讨论了吸引边界相互作用的非渐近情形，并证明了如果区域是半平面或角小于$π$的楔形体的局部变形，则离散谱是非空的;如果区域是凹的，则离散谱是空的。
英文摘要：
We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a two-term asymptotic formula in which the next-to-leading term is determined by the extremum of the boundary curvature. We also discuss the non-asymptotic case of attractive boundary interaction and show that the discrete spectrum is nonempty if the domain is a local deformation of a halfplane or a wedge of angle less than $π$, and it is void if the domain is concave.
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