Quasinormal modes of a massless charged scalar field on a small Reissner-Nordström-anti-de Sitter black hole

We investigate quasinormal modes of a massless charged scalar field on a small Reissner-Nordstr\"om-anti-de Sitter (RN-AdS) black hole both with analytical and numerical approaches. In the analytical approach, by using the small black hole approximation (${r}_{+}\ensuremath{\ll}L$), we obtain the quasinormal mode frequencies in the limit of ${r}_{+}/L\ensuremath{\rightarrow}0$, where ${r}_{+}$ and $L$ stand for the black hole event horizon radius and the AdS scale, respectively. We then show that the small RN-AdS black hole is unstable if its quasinormal modes satisfy the superradiance condition and that the instability condition of the RN-AdS black hole in the limit of ${r}_{+}/L\ensuremath{\rightarrow}0$ is given by $Q>\frac{3}{eL}{Q}_{c}$, where $Q$, ${Q}_{c}$, and $e$ are the charge of the black hole, the critical (maximum) charge of the black hole, and the charge of the scalar field, respectively. In the numerical approach, we calculate the quasinormal modes for the small RN-AdS black holes with ${r}_{+}\ensuremath{\ll}L$ and confirm that the RN-AdS black hole is unstable if its quasinormal modes satisfy the superradiance condition. Our numerical results show that the RN-AdS black holes with ${r}_{+}=0.2L$, $0.1L$, and $0.01L$ become unstable against scalar perturbations with $eL=4$ when the charge of the black hole satisfies $Q\ensuremath{\gtrsim}0.8{Q}_{c}$, $0.78{Q}_{c}$, and $0.76{Q}_{c}$, respectively.