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The object of this paper is the generalization of the theoretical calculation of the probability density of finding a charged particle crossing the double  arc -slit in the presence of the vector potential. The explicit derivations is carried out in cylindrical coordinates for the Green functions for the underlying problem corresponding to a particle in free magnetic field, for a quantized as well as a non-quantized flux. The discussion in this paper will be limited to the uniform wavefunction of each particle on the surfaces of double arc-slits.

Three dimensional Green function in cylindrical coordinates: application to the Aharonov–Bohm effect of double arc-slits