Tuning A Rotating Black Hole Spectrum with Dark Matter Halo: Quasibound States, Scalar Cloud, Black Hole Bomb and Superradiant Scattering

Abstract

We investigate the spectral dynamics of a rotating black hole embedded in a Dehnen $(1,4,\gamma)$ dark matter halo, where quasibound states and superradiant scattering jointly characterize the physical response of the system. Starting from an exact Schwarzschild--Dehnen solution, we construct its rotating counterpart via the Newman--Janis algorithm, yielding a consistent axisymmetric geometry that incorporates the influence of a structured halo. The Dehnen profile, through its inner slope parameter $\gamma$, introduces a controlled deformation of both the near-horizon and asymptotic regions of the spacetime. Using the analytical asymptotic matching method, we derive the quasibound-state spectrum and show that the real part of the frequency retains a hydrogen-like structure, but is systematically shifted by the halo through the effective mass scale $\rho_0 r_0^3/(\gamma-3)$. In particular, denser, more extended, and more cuspy halos enhance the binding energy, lower the critical mass required for the onset of instability, and typically suppress the growth rate of the black hole bomb. In the scattering sector, we obtain an analytic expression for the superradiant amplification factor and find that the same halo properties that strengthen binding effects also tend to narrow the superradiant window. These results demonstrate that quasibound states and superradiant scattering are complementary manifestations of a unified spectral structure, with the Dehnen halo acting as an environmental tuner that imprints its properties directly onto both the resonance spectrum and the energy-extraction channels of the rotating black hole.