Two types of quasinormal modes of Casadio-Fabbri-Mazzacurati brane-world black holes

Using the convergent Leaver method, we investigate the quasinormal modes of a massive scalar field propagating in the background of the Casadio–Fabbri–Mazzacurati (CFM) brane-world black hole. We show that the spectrum exhibits two distinct types of modes, depending on their behavior as the field mass increases. In one class, the real oscillation frequency decreases and eventually approaches zero, while in the other the damping rate tends to vanish. When either the real or imaginary part of the frequency reaches zero, the corresponding mode disappears from the spectrum, and the first overtone replaces it. The emergence of modes with a vanishing real part at certain critical values of the field mass is a distinctive feature of the CFM spectrum.