The extremal Reissner-Nordstr\"om black holes: an exact charged scalar quasiresonance

In this letter, we present a novel exact scalar quasibound states solutions in the extremal Reissner-Norstr\"om black hole background. We start with the construction of the governing covariant relativistic scalar field equation, the Klein-Gordon equation in the extremal Reissner-Norstr\"om black hole background and applying the separation of variables anzat. The exact relativistic scalar wave's angular solution is found in terms of the spherical harmonics while the two independent radial wave solutions are, for the first time, exactly found and presented in terms of the double confluent Heun functions. The solutions are settled in the gravitational potential well and behave like an ingoing waves approaching black hole's horizon, vanishing when approaching infinity. The gravitationally bounded charged massive scalar fields are found to have quantized complex valued energy levels while imaginary energy levels are obtained for the charged massless scalar field, of both cases, indicating decaying states. Further investigation shows that the extreme Reissner-Nordst\"om black hole does not support scalar cloud. And with the help of the obtained exact radial solutions, the Hawking radiation of the extremal Reissner-Nordst\"om black hole is investigated and we find the zero temperature of the black hole's horizon.