We consider a class of two-dimensional Schrödinger operator with a singular interaction of the $$\delta $$δ type and a fixed strength $$\beta $$β supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov–Bohm flux $$\alpha \in [0,\frac{1}{2}]$$α∈[0,12] in the center. It is shown that if $$\beta \ne 0$$β≠0, there is a critical value $$\alpha _{\mathrm {crit}}\in (0,\frac{1}{2})$$αcrit∈(0,12) such that the discrete spectrum has an accumulation point when $$\alpha <\alpha _{\mathrm {crit}}$$α<αcrit, while for $$\alpha \ge \alpha _{\mathrm {crit}}$$α≥αcrit the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $$\alpha \in (0,\frac{1}{2})$$α∈(0,12) and $$|\beta |$$|β| small enough.

中文翻译：

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Aharonov 和 Bohm 与威尔士特征值

我们考虑一类二维薛定谔算子，它具有 $$\delta $$δ 类型和固定强度 $$\beta $$β 的奇异相互作用，由无限的同心、等距间隔的圆族支持，并讨论什么当系统被中心的 Aharonov-Bohm 通量 $$\alpha \in [0,\frac{1}{2}]$$α∈[0,12] 修正时，发生在基本谱以下。表明如果$$\beta \ne 0$$β≠0，则存在一个临界值$$\alpha _{\mathrm {crit}}\in (0,\frac{1}{2})$ $αcrit∈(0,12)使得离散谱在$$\alpha <\alpha _{\mathrm {crit}}$$α<αcrit时有一个累积点，而对于$$\alpha \ge \alpha _ {\mathrm {crit}}$$α≥αcrit 特征值的数量至多是有限的，特别是对于任何固定的$$\alpha \in (0,\frac{1}{2})，离散谱都是空的$$α∈(0,12) 和 $$|\beta |$$|β| 足够小。