Spontaneous scalarization of charged Reissner-Nordstrom 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 is an element of{[alpha(-)(n) ((Q) over bar, alpha(n)(+)((Q) over bar]}(n=0)(n=infinity), where alpha is the dimensionless coupling constant of the theory and (Q) over bar Q/Mis the dimensionless charge-to-mass ratio of the central supporting black hole. In the present paper we use analyticaltechniques 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-Nordstrom black holes. In particular, we derive a remarkably compact formula for the discrete resonant spectrum {a(n)(-)((Q) over bar)}(n=0)(n=infinity) which characterizes the composed black-hole-linearized-field configurations along the existence-lineof the system, the critical line which separates bare Reissner-Nordstrom black holes from hairy scalarized black-hole configurations. The analytical results are confirmed by direct numerical computations. (C) 2019 The Author(s). Published by Elsevier B.V.