We investigate the spectrum of a soft quantum waveguide in two dimensions of the generalized ‘bookcover’ shape, that is, Schrödinger operator with the potential in the form of a ditch consisting of a finite curved part and straight asymptotes which are parallel or almost parallel pointing in the same direction. We show how the eigenvalues accumulate when the angle between the asymptotes tends to zero. In case of parallel asymptotes the existence of a discrete spectrum depends on the ditch profile. We prove that it is absent in the weak-coupling case, on the other hand, it exists provided the transverse potential is strong enough. We also present a numerical example in which the critical strength can be assessed.

中文翻译：

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软波导中的隧道：合上一本书

我们研究了广义“书皮”形状二维软量子波导的光谱，即薛定谔算子，其势能呈沟渠形式，由有限弯曲部分和平行或几乎平行指向的直渐近线组成朝同一方向。我们展示了当渐近线之间的角度趋于零时特征值如何累积。在平行渐近线的情况下，离散谱的存在取决于沟渠剖面。我们证明在弱耦合情况下它不存在，另一方面，只要横向电势足够强，它就存在。我们还提供了一个可以评估临界强度的数值示例。