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