  It has recently been demonstrated that static spatially regular scalar
fields, which are non-minimally coupled to the electromagnetic field of a
charged central black hole, can be supported in the exterior regions of the
black-hole spacetime. In the present paper we use {\it analytical} techniques
in order to study the physical and mathematical properties of the externally
supported linearized scalar field configurations (scalar 'clouds') in the
dimensionless large-mass regime $\mu r_+\gg1$ (here $\mu$ and $r_+$ are
respectively the proper mass of the supported scalar field and the outer
horizon radius of the central supporting black hole). In particular, we derive
a remarkably compact analytical formula for the discrete resonant spectrum
$\{\alpha_n(\mu;Q/M)\}_{n=0}^{n=\infty}$ which characterizes the dimensionless
coupling parameter of the composed
black-hole-non-minimally-coupled-linearized-massive-scalar-field
configurations. The physical significance of this resonant spectrum stems from
the fact that, for a given value of the dimensionless black-hole electric
charge $Q/M$, the fundamental (smallest) eigenvalue $\alpha_0(\mu)$ determines
the critical existence-line of the composed black-hole-massive-field system, a
boundary line which separates non-linearly coupled hairy
charged-black-hole-massive-scalar-field configurations from bald
Reissner-Nordstr\"om black holes. The analytical results derived in this paper
are confirmed by direct numerical computations.
