Persistent currents for 2D Schrödinger operator with a strong $δ$-interaction on a loop

Abstract

We investigate the two-dimensional magnetic Schrödinger operator $H_{B,β}=(-i\nabla-A)^2 -βδ(\cdot-Γ)$, where $Γ$ is a smooth loop and the vector potential $A$ corresponds to a homogeneous magnetic field $B$ perpendicular to the plane. The asymptotics of negative eigenvalues of $H_{B,β}$ for $β\to\infty$ is found. It shows, in particular, that for large enough positive $β$ the system exhibits persistent currents.