Can a repulsive potential in graphene have boundstates in a magnetic field?

We consider Klein tunneling through a repulsive and cylindrical potential with range R and strength V . Recently it was found that, in the strong coupling regime R/ℓ <1, the repulsive potential can have bound states peaked inside the potential with tails extending over ℓ2(N +1), where N is Landau level (LL) index and ℓ is the magnetic length. The presence of these bound states is a consequence of a subtle interplay between Klein tunneling and quantization effect of magnetic ?elds. Because of the presence of these bound states the effective coupling between the repulsive potential and an electron can be attractive. Here we show that this effect is a consequence of singular interaction between the repulsive potential and an electron that cannot be captured in perturbative approaches.