Abstract

The paper reviews spectral properties of a class of singular Schrödinger operators with the interaction supported by manifolds or complexes of codimension \(1\); in particular, the relation of these properties to the geometric setting of the problem is discussed. We describe how these operators can be approximated by operators of other classes and how such approximations can be used. Furthermore, we present asymptotic expansions of the eigenvalues in terms of the parameters characterizing the coupling strength and geometric deformations. We also give an example illustrating the influence of a magnetic field of the Aharonov–Bohm type and briefly describe results on singular perturbations of Dirac operators.

中文翻译：

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泄漏的量子结构

摘要

本文综述了一类奇异Schrödinger算子的谱性质，该奇异Schrödinger算子具有余维\（1 \）的流形或复数支持的相互作用; 特别是，讨论了这些属性与问题的几何设置之间的关系。我们描述了如何用其他类的运算符来近似这些运算符，以及如何使用这种近似值。此外，根据表征耦合强度和几何变形的参数，我们给出了特征值的渐近展开。我们还提供了一个示例，说明了Aharonov–Bohm型磁场的影响，并简要描述了Dirac算子奇异摄动的结果。