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We consider the Aharonov-Bohm magnetic potential and study the transitionfrom normal to superconducting solutions within the Ginzburg-Landau model ofsuperconductivity. We obtain oscillations consistent with the Little-Parkseffect. We study the same problem but for a regularization of the Aharonov-Bohmpotential, which leads to an interesting Aharonov-Bohm like magnetic field, andwe prove that the transition between superconducting and normal solutions isnot monotone too. Our results show a mechanism to derive the Aharonov-Bohmmagnetic potential starting from a step magnetic field thereby presenting a newaspect of magnetic steps, besides their favoring of the celebrated edge states.

Oscillatory patterns in the Ginzburg-Landau model driven by the  Aharonov-Bohm potential (Derivation of the Aharanov-Bohm potential in the  Ginzburg-Landau model)