Aharonov and Bohm vs. Welsh eigenvalues

We consider a class of two-dimensional Schr"odinger operator with a singular interaction of the $\delta$ type and a fixed strength $\beta$ 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 $\alpha \in [0,\frac{1}{2}]$ in the center. It is shown that if $\beta \mathrm{\ne }0$, there is a critical value ${\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\in (0,\frac{1}{2})$ such that the discrete spectrum has an accumulation point when $\alpha <{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, while for $\alpha \ge {\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $\alpha \in (0,\frac{1}{2})$ and $\mathrm{\mid }\beta \mathrm{\mid }$ small enough.