General Heun function, Damour–Ruffini method and Klein–Gordon equation in static black holes with nonlinear electrodynamics

We investigate the dynamics of a scalar field in the spacetime of a static, spherically symmetric black hole arising from non-linear Einstein–Maxwell theory, which introduces rich structures beyond the standard Reissner–Nordström solution. By solving the covariant Klein–Gordon equation, we obtain novel exact radial solutions expressed through General Heun functions. Imposing the associated polynomial condition leads to a fully analytical, quantized energy spectrum, providing an exact formula for the quasibound state eigenenergies. Moreover, employing the Damour–Ruffini method, we study Hawking radiation and derive an explicit expression for the Hawking temperature at the event horizon. These results demonstrate how non-linear electrodynamic effects modify scalar field dynamics and black hole thermodynamics, offering a new framework to explore quantum phenomena in strongly curved spacetimes.