Aharonov and Bohm vs. Welsh eigenvalues

TL;DR

This paper investigates how an Aharonov-Bohm flux affects the discrete spectrum of a two-dimensional Schrödinger operator with delta interactions on concentric circles, revealing a critical flux value that determines the spectrum's accumulation.

Contribution

It introduces a critical flux parameter for the spectral behavior of Schrödinger operators with delta interactions under Aharonov-Bohm flux, showing how the discrete spectrum is influenced.

Findings

Existence of a critical flux value $\alpha_{crit}$ affecting eigenvalue accumulation.

Discrete spectrum becomes finite or empty depending on the flux and interaction strength.

For small interaction strength, the spectrum is empty for all flux values above zero.

Abstract

We consider a class of two-dimensional Schr\"odinger operator with a singular interaction of the $\delta$ type and a fixed strength $\beta$ 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 $\alpha\in [0,\frac12]$ in the center. It is shown that if $\beta\ne 0$, there is a critical value $\alpha_\mathrm{crit} \in(0,\frac12)$ such that the discrete spectrum has an accumulation point when $\alpha<\alpha_\mathrm{crit} $, while for $\alpha\ge\alpha_\mathrm{crit} $ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $\alpha\in (0,\frac12)$ and $|\beta|$ small enough.