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It has recently been demonstrated that charged black holes can supportspatially regular matter configurations made of massless scalar fields whichare non-minimally coupled to the electromagnetic field of the chargedspacetime. Intriguingly, using numerical techniques, it has been revealed thatthe resonant spectra of the composedcharged-black-hole-nonminimally-coupled-scalar-field configurations arecharacterized by charge-dependent discrete scalarization bands$\alpha\in\{[\alpha^{-}_{n}({\bar Q}),\alpha^{+}_{n}({\barQ}]\}_{n=0}^{n=\infty}$, where $\alpha$ is the dimensionless coupling constantof the theory and ${\bar Q}\equiv Q/M$ is the dimensionless charge-to-massratio of the central supporting black hole. In the present paper we use {\itanalytical} techniques in order to study the physical and mathematicalproperties of the spatially regular non-minimally coupled scalar fieldconfigurations (linearized scalar clouds) which are supported by the centralcharged Reissner-Nordstr\"om black holes. In particular, we derive a remarkablycompact formula for the discrete resonant spectrum $\{\alpha^-_n({\barQ})\}_{n=0}^{n=\infty}$ which characterizes the composedblack-hole-linearized-field configurations along the {\it existence-line} ofthe system, the critical line which separates bare Reissner-Nordstr\"om blackholes from hairy scalarized black-hole configurations. The analytical resultsare confirmed by direct numerical computations.

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