Stability analysis of a charged black hole with a nonlinear complex scalar field

It has been shown recently that the charged black hole can be scalarized if Maxwell fieldminimally couples with a complex scalar which has nonnegative nonlinear potential. We first prove that such scalarization cannot be a result of continuous phase transition for general scalar potential. Furthermore, we numerically find that it is possible that the RN black hole will be scalarized by a first order phase transition spontaneously and near extremal Reissner-Nordstrom black hole(RN black hole) is not stable in the microcanonical ensemble. In addition, considering a massless scalar perturbation, we compute the quasinormal modes of the scalarized charged black hole and the results do not only imply that the spontaneously scalarized charged black hole is favored in thermodynamics but also suggest that it is kinetically stable against scalar perturbation at linear level. Our numerical results also definitely gives negative answer to Penrose-Gibbons conjecture and two new versions of Penrose inequality in charged case are suggested.