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

Abstract

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