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

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