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It has recently been demonstrated that asymptotically flat neutral reflectingstars are characterized by an intriguing no-hair property. In particular, ithas been proved that these {\it horizonless} compact objects cannot supportspatially regular {\it static} matter configurations made of scalar (spin-0)fields, vector (spin-1) fields, and tensor (spin-2) fields. In the presentpaper we shall explicitly prove that spherically symmetric compact reflectingstars can support {\it stationary} (rather than static) bound-state massivescalar fields in their exterior spacetime regions. To this end, we solveanalytically the Klein-Gordon wave equation for a linearized scalar field ofmass $\mu$ and proper frequency $\omega$ in the curved background of aspherically symmetric compact reflecting star of mass $M$ and radius$R_{\text{s}}$. It is proved that the regime of existence of these stationarycomposed star-field configurations is characterized by the simple inequalities$1-2M/R_{\text{s}}<(\omega/\mu)^2<1$. Interestingly, in the regime$M/R_{\text{s}}\ll1$ of weakly self-gravitating stars we derive a remarkablycompact {\it analytical} formula for the discrete spectrum$\{\omega(M,R_{\text{s}},\mu)\}^{n=\infty}_{n=1}$ of resonant oscillationfrequencies which characterize the stationary composedcompact-reflecting-star-linearized-massive-scalar-field configurations.Finally, we verify the accuracy of the analytically derived resonance formulaof the composed star-field configurations with direct numerical computations.

Stationary bound-state massive scalar field configurations supported by spherically symmetric compact reflecting stars