Abstract
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 ∈ [0, ½] in the center. It is shown that if β ≠ 0 , there is a critical value αcrit ∈ (0, ½) such that the discrete spectrum has an accumulation point when α < αcrit , while for α ≥ αcrit the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed α ∈ (0, ½) and |β| small enough.