Aharonov and Bohm versus Welsh eigenvalues

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∈[0,12] 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∈(0,12) and | β| small enough.