表面量子带
Quantum strips on surfaces
摘要：
受量子波导理论的启发，我们研究了拉普拉斯算子的频谱，受Dirichlet边界条件，在一个弯曲的带的恒定宽度，被定义为一个管状的邻域中的一个无限的曲线在二维黎曼流形。在适当的假设下，带是渐近直的，我们本地化的本质谱，并找到充分条件，保证几何诱导的束缚态的存在。特别是，离散谱存在的非负弯曲带进行了详细的研究。一般的结果被用来恢复和重温已知的事实，量子带在平面上。作为非正弯曲量子带的一个例子，我们考虑直纹表面上的带。
英文摘要：
Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a two-dimensional Riemannian manifold. Under the assumption that the strip is asymptotically straight in a suitable sense, we localise the essential spectrum and find sufficient conditions which guarantee the existence of geometrically induced bound states. In particular, the discrete spectrum exists for non-negatively curved strips which are studied in detail. The general results are used to recover and revisit the known facts about quantum strips in the plane. As an example of non-positively curved quantum strips, we consider strips on ruled surfaces.
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