Aharonov和Bohm vs. Welsh特征值
Aharonov and Bohm vs. Welsh eigenvalues
摘要：
我们考虑一类二维薛定谔算子，其奇异相互作用为$δ$型，且$β$强度固定，由无限族同心等距圆支撑，并讨论了当系统被中心的Aharonov-Bohm通量$α\in [0,\frac12]$修正时，在本质谱以下会发生什么。证明了当$β\ne 0$为$α<α_\mathrm{crit} $时，存在一个临界值$α_\mathrm{crit} \in(0,\frac12)$，使得离散谱有一个聚点，而对于$α\geα_\mathrm{crit} $，特征值的个数至多是有限的，特别地，对于任意固定的$α\in (0,\frac12)$和足够小的$|β|$，离散谱为空.
英文摘要：
We consider a class of two-dimensional Schrödinger operator with a singular interaction of the $δ$ type and a fixed strength $β$ 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 $α\in [0,\frac12]$ in the center. It is shown that if $β\ne 0$, there is a critical value $α_\mathrm{crit} \in(0,\frac12)$ such that the discrete spectrum has an accumulation point when $α<α_\mathrm{crit} $, while for $α\geα_\mathrm{crit} $ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $α\in (0,\frac12)$ and $|β|$ small enough.
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