Stationary scalar configurations around extremal charged black holes

We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordstr\"om background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies \omega for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies $\mathcal{R}(\omega)$ tends to the mass of the field and the imaginary part $\mathcal{I}(\omega)$ tends to zero, for any angular momentum quantum number $\ell$. The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system.