Asymptotics of eigenvalues of the operator describing Aharonov-Bohm effect combined with homogeneous magneticfield coupled with a strong $\delta$-interaction on a loop

Abstract

We investigate the two-dimensional magnetic operator $H_{c_0,B,\beta }={(-i\nabla -A)}^2-\beta \delta (.-\Gamma ),$ where $\Gamma$ is a smooth loop. The vector potential has the form $A=c_0(\frac{-y}{x^2+y^2};\frac{x}{x^2+y^2})+\frac{B}{2}(-y;x)$; $B>0,$ $c_0\in ]0;1[$. The asymptotics of negative eigenvalues of $H_{c_0,B,\beta }$ for $\beta ⟶+\infty$ is found. We also prove that for large enough positive value of $\beta$ the system exhibits persistent currents.

11 pages