Infinitesimally thin static scalar shells surrounding charged Gauss-Bonnet black holes

We reveal the existence of a new form of spontaneously scalarized black-hole configurations. In particular, it is proved that Reissner-Nordstr\"om black holes in the highly charged regime $Q/M>(Q/M)_{\text{crit}}=\sqrt{21}/5$ can support {\it thin} matter shells that are made of massive scalar fields with a non-minimal coupling to the Gauss-Bonnet invariant of the curved spacetime. These static scalar shells, which become infinitesimally thin in the dimensionless large-mass $M\mu\gg1$ regime, hover a finite proper distance above the black-hole horizon [here $\{M,Q\}$ are respectively the mass and electric charge of the central supporting black hole, and $\mu$ is the proper mass of the supported scalar field]. In addition, we derive a remarkably compact analytical formula for the discrete resonance spectrum $\{\eta(Q/M,M\mu;n)\}_{n=0}^{n=\infty}$ of the non-trivial coupling parameter which characterizes the bound-state charged-black-hole-thin-massive-scalar-shell cloudy configurations of the composed Einstein-Maxwell-scalar field theory.