  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.
