This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter $h > 0$ in the case when the magnetic field vanishes along a smooth curvewhich crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit $h \to 0$. We show that each crossing point acts as a potential well, generating a new decay scale of $h^{3/2}$ for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in $\mathbb R^2$ for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0.

中文翻译：

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关于具有自相交零集的磁场的半经典拉普拉斯算子

本文致力于在磁场沿光滑曲线消失并在有界区域内交叉的情况下，对具有半经典参数$ h> 0 $的二维磁性拉普拉斯算子的诺伊曼实现进行频谱分析。我们研究其特征对在$ h \至0 $的极限下的行为。我们表明，每个交叉点都充当势阱，从而为最低特征值生成新的衰变标度$ h ^ {3/2} $，并为交叉点集周围的特征向量生成指数集中。这些性质是$ \ mathbb R ^ 2 $中相关模型问题的性质的结果，其磁场的零集是两条直线的并集。在本文中，我们还分析了当两条直线之间的角度趋于0时模型问题的范围。