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 $\alpha\in [0,\frac12]$ in the center. It is shown that if $\beta\ne 0$, there is a critical value $\alpha_\mathrm{crit} \in(0,\frac12)$ such that the discrete spectrum has an accumulation point when $\alpha<\alpha_\mathrm{crit}$, while for $\alpha\ge\alpha_\mathrm{crit}$ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $\alpha\in (0,\frac12)$ and $|\beta|$ small enough.