Spontaneous scalarization of charged Reissner-Nordstr\"om black holes: Analytic treatment along the existence line

It has recently been demonstrated that charged black holes can support spatially regular matter configurations made of massless scalar fields which are non-minimally coupled to the electromagnetic field of the charged spacetime. Intriguingly, using numerical techniques, it has been revealed that the resonant spectra of the composed charged-black-hole-nonminimally-coupled-scalar-field configurations are characterized by charge-dependent discrete scalarization bands $\alpha\in\{[\alpha^{-}_{n}({\bar Q}),\alpha^{+}_{n}({\bar Q}]\}_{n=0}^{n=\infty}$, where $\alpha$ is the dimensionless coupling constant of the theory and ${\bar Q}\equiv Q/M$ is the dimensionless charge-to-mass ratio of the central supporting black hole. In the present paper we use {\it analytical} techniques in order to study the physical and mathematical properties of the spatially regular non-minimally coupled scalar field configurations (linearized scalar clouds) which are supported by the central charged Reissner-Nordstr\"om black holes. In particular, we derive a remarkably compact formula for the discrete resonant spectrum $\{\alpha^-_n({\bar Q})\}_{n=0}^{n=\infty}$ which characterizes the composed black-hole-linearized-field configurations along the {\it existence-line} of the system, the critical line which separates bare Reissner-Nordstr\"om black holes from hairy scalarized black-hole configurations. The analytical results are confirmed by direct numerical computations.