Regular scalar clouds around a Kerr-Newman black hole: subextremal and extremal scenarios

In this work we analyze the existence of electrically charged scalar clouds which are bound states of a complex-valued massive scalar field in the background of subextremal and extremal Kerr-Newman black holes (BH). In particular, we reanalyze neutral (uncharged) clouds in extremal Kerr BH's. For the extremal scenarios we have implemented a novel technique which allows us to obtain regular clouds at the BH horizon ${\cal H}^+$ which turn out to be connected ``continuously'' with the cloud solutions in the subextremal case even if some derivatives of the scalar field are unbounded at the horizon. In particular, for subextremal BH's we have established regularity conditions at ${\cal H}^+$, by demanding that the field and its radial derivatives are bounded there, but in the extremal scenarios we relax this last condition while demanding that some scalar invariants are well behaved at ${\cal H}^+$. Furthermore we have implemented an integral technique to understand and justify in a simple and heuristic way the existence of such cloud configurations in those BH backgrounds.