Asymptotics of eigenvalues of the Aharonov-Bohm operator with a strong $\delta$-interaction on a loop

TL;DR

This paper analyzes the asymptotic behavior of negative eigenvalues of a two-dimensional Aharonov-Bohm operator with a strong delta interaction on a loop, revealing their growth and the emergence of persistent currents for large interaction strength.

Contribution

It provides the first detailed asymptotic analysis of eigenvalues for this operator with a strong delta interaction on a loop, and demonstrates the existence of persistent currents.

Findings

Negative eigenvalues grow asymptotically as the interaction strength increases.

Persistent currents occur for sufficiently large positive delta interaction.

The asymptotic behavior of eigenvalues is explicitly characterized.

Abstract

We investigate the two-dimensional Aharonov-Bohm operator $H_{c_0,\beta} = {(-i\nabla -A)}^{2}-\beta\delta(.-\Gamma),$ where $\Gamma$ is a smooth loop and $A$ is the vector potential which corresponds to Aharonov-Bohm potential. The asymptotics of negative eigenvalues of $H_{c_0,\beta}$ for $\beta \longrightarrow +\infty$ is found. We also prove that for large enough positive value of $\beta$ the system exhibits persistent currents.