Charged reflecting stars supporting charged massive scalar field configurations

The recently published no-hair theorems of Hod, Bhattacharjee, and Sarkar have revealed the intriguing fact that horizonless compact reflecting stars {\it cannot} support spatially regular configurations made of scalar, vector and tensor fields. In the present paper we explicitly prove that the interesting no-hair behavior observed in these studies is not a generic feature of compact reflecting stars. In particular, we shall prove that charged reflecting stars {\it can} support {\it charged} massive scalar field configurations in their exterior spacetime regions. To this end, we solve analytically the characteristic Klein-Gordon wave equation for a linearized charged scalar field of mass $\mu$, charge coupling constant $q$, and spherical harmonic index $l$ in the background of a spherically symmetric compact reflecting star of mass $M$, electric charge $Q$, and radius $R_{\text{s}}\gg M,Q$. Interestingly, it is proved that the discrete set $\{R_{\text{s}}(M,Q,\mu,q,l;n)\}^{n=\infty}_{n=1}$ of star radii that can support the charged massive scalar field configurations is determined by the characteristic zeroes of the confluent hypergeometric function. Following this simple observation, we derive a remarkably compact analytical formula for the discrete spectrum of star radii in the intermediate regime $M\ll R_{\text{s}}\ll 1/\mu$. The analytically derived resonance spectrum is confirmed by direct numerical computations.