Plummer Dark Matter Black Hole with Topological Defects: Shadow, Greybody Factors, Quasinormal Modes, and Thermodynamics

We investigate a static, spherically symmetric black hole spacetime embedded in a cored Plummer dark matter halo and surrounded by a cloud of strings. The metric is shown to arise as an exact solution of the Einstein field equations sourced by a two-component anisotropic fluid whose energy-momentum tensor decomposes additively into a Plummer-halo part and a Letelier string-cloud part, and the resulting matter content is verified, both symbolically with Maple and numerically over the relevant parameter range, to satisfy all four standard pointwise energy conditions (null, weak, strong, and dominant) throughout the exterior region. We analyze its optical and dynamical properties by computing the photon sphere and shadow radii, as well as the weak deflection angle using the Gauss-Bonnet theorem. The innermost stable circular orbit is also determined. The shadow predictions are then confronted with the Event Horizon Telescope measurements of M87$^{*}$ and Sgr~A$^{*}$, yielding parameter bounds on the cloud-of-strings tension and the Plummer halo density. Scalar perturbations are studied through the effective potential, greybody factor bounds via the Boonserm-Visser method, the Hawking radiation spectrum, and quasinormal mode frequencies using the WKB approximation. A thermodynamic analysis is performed, including the Hawking temperature, entropy, heat capacity, Smarr-like relation, and Gibbs free energy. The heat capacity remains negative for all parameter ranges, indicating thermodynamic instability and the absence of any Davies-type phase transition. Overall, a clear hierarchy emerges: the string cloud parameter provides the dominant corrections to all observables, while the Plummer halo density contributes subdominant effects.