Abstract: In this talk we discuss a class of two-dimensional Schrödinger operator with a singular interaction of the $\delta$ type and a fixed strength supported by an infinite family of concentric, equidistantly spaced circles. We analyze what happens below the essential spectrum after implementing an Aharonov-Bohm flux $\alpha \in [0,1/2]$ in the center. We prove that there exists a critical value $\alpha_{\mathrm{cr}} \in (0, 1/2)$ such that the discrete spectrum has an accumulation point when $\alpha < \alpha_{\mathrm{cr}}$, while for $\alpha \geq \alpha_{\mathrm{crit}}$ the number of eigenvalues is finite. The talk is based on a common work with P. Exner.