Pavel Exner and Sylwia Kondej
Aharonov and Bohm vs. Welsh eigenvalues
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ABSTRACT. We consider a class of two-dimensional Schr\"odinger operator with a singular interaction of the $\delta$ type and a fixed strength $eta$ 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 $lpha\in [0,rac12]$ in the center. It is shown that if $eta
e 0$, there is a critical value $lpha_\mathrm{crit}\in(0,rac12)$ such that the discrete spectrum has an accumulation point when $lpha<lpha_\mathrm{crit} $, while for $lpha\gelpha_\mathrm{crit} $ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $lpha\in (0,rac12)$ and $|eta|$ small enough.