Aharonov and Bohm vs. Welsh eigenvalues

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, 1 2 ] in the center. It is shown that if β ̸= 0, there is a critical value αcrit ∈ (0, 12) 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, 1 2 ) and |β| small enough. Mathematics Subject Classification (2010). 81Q10, 35J10.