Persistent currents for 2D Schr\"odinger operator with a strong \(\delta\)-interaction on a loop

Abstract

We investigate the two-dimensional magnetic Schr\"odinger operator \(H_{B,\beta}=(-i\nabla-A)^2 -\beta\delta(\cdot-\Gamma)\), where \(\Gamma\) 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 \(H_{B,\beta}\) for \(\beta\to\infty\) is found. It shows, in particular, that for large enough positive \(\beta\) the system exhibits persistent currents.