Ringdown of a black hole sourced by a Burkert-density effective anisotropic source

We construct a static, spherically symmetric black hole spacetime sourced by an effective anisotropic matter distribution whose energy density follows the cored Burkert profile. The source should be understood as a Burkert-density effective fluid, rather than as a microscopic model of pressureless collisionless dark matter. Solving the Einstein equations under this closure condition, we obtain an analytic Schwarzschild-like metric that reduces smoothly to the vacuum Schwarzschild solution when the halo contribution vanishes. We then study axial gravitational perturbations of this geometry and determine the associated quasinormal spectrum using complementary frequency-domain and time-domain methods. We find that increasing either the core radius r0 or the central density ρ0 shifts the ringdown toward lower frequency and weaker damping, with the effect of r0 being more pronounced. The close agreement among the numerical extractions supports the reliability of the results. Our analysis provides a useful benchmark for assessing how a cored Burkert-type effective environment can modify black hole ringdown.