Stability of scalarized charged black holes in the Einstein-Maxwell-Scalar theory

We analyze the stability of scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory with quadratic coupling. These black holes are labelled by the number of n = 0, 1, 2,..., where n = 0 is called the fundamental black hole and n = 1, 2,... denote the n-excited black holes. We show that the n = 0 black hole is stable against full perturbations, whereas the n = 1, 2 excited black holes are unstable against the s(l = 0)-mode scalar perturbation. This is consistent with the EMS theory with exponential coupling, but it contrasts to the n = 0 scalarized black hole in the Einstein-Gauss-Bonnet-Scalar theorywith quadratic coupling. This implies that the endpoint of unstable Reissner-Nordstrom black holes with alpha > 8.019 is the n = 0 black hole with the same q. Furthermore, we study the scalarized charged black holes in the EMS theory with scalar mass m(phi)(2) f = alpha/beta.