Quasinormal modes of hairy black holes with mixed couplings

In this paper, we study a mixed coupling function incorporating spontaneous and non-linear scalarization of the hairy black hole in the Einstein-Maxwell-scalar model. Recent researches have demonstrated that a series of hairy black holes exhibit multiple photon spheres outside the horizon. Compared to the spontaneous solution, the mixed coupling function leads to a smaller event horizon radius. Quasinormal modes describe the fundamental signal of the gravitational waves in the ringdown stage, providing an effective method for detection. The real part of quasinormal modes grows with the azimuthal quantum number, suggesting the positive correlation between quasinormal modes and the angular velocity of trajectories. Furthermore, the absolute value of quasinormal frequencies' imaginary part is connected to the Lyapunov exponents of the corresponding photon spheres. The result of quasinormal modes calculated by the WKB approximation indicates long-lived and sub-long-lived modes, which are associated with the stable and the outer unstable photon spheres.