Quasinormal modes of scalarized black holes in the Einstein-Maxwell-Scalar theory

We perform the stability analysis on scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory by computing quasinormal mode spectrum. It is noted that the appearance of these black holes with scalar hair is closely related to the instability of Reissner-Nordstr"om black holes without scalar hair in the EMS theory. The scalarized black hole solutions are classified by the node number of $n=0,1,2,\cdots$, where $n=0$ is called the fundamental branch and $n=1,2,\cdots$ denote the $n$ excited branches. Here, we show that the $n=1,2$ excited black holes are unstable against against the $s(l=0)$-mode scalar perturbation, while the $n=0$ fundamental black hole is stable against all scalar-vector-tensor perturbations. This is consistent with other scalarized black holes without charge found in the Einstein-Scalar-Gauss-Bonnet theory.