Schwarzschild Black Hole Surrounded by Perfect Fluid Dark Matter in the presence of Quintessence Matter Field

In this paper, we present an exact solution for a spherically symmetric Schwarzschild black hole surrounded by perfect fluid dark matter (PFDM) in the presence of a quintessence field. We investigate the impact of dark matter on the black hole's thermodynamic and optical properties, as well as its quasinormal modes. Our analysis reveals a critical radius where the heat capacity becomes positive, indicating the thermodynamic stability of the black hole. Notably, this critical radius increases with the dark matter parameter $\alpha$. Additionally, we find that as the effects of dark matter increase, the black hole's shadow radius decreases. Using the WKB approximation, we show that the quasinormal mode spectrum differs from that of a standard Schwarzschild black hole due to the influence of PFDM. Moreover, we demonstrate that as $\alpha$ increases, both the real part and the magnitude of the imaginary part of the quasinormal mode frequencies increase. This suggests that field perturbations decay more rapidly in the presence of PFDM compared to those in a Schwarzschild black hole.