Abstract: Reasonable approximations are introduced to investigate the real scalar field scattering in the nearly extremal Schwarzschild–de Sitter (SdS) space. The approximations naturally lead to the invertible x(r) and the global replacement of the true potential by a P?shl–Teller one. Meanwhile, the Schr$\ddot{\rm o}$dinger-like wave equation is transformed into a solvable form. Our numerical solutions to the wave equation show that the wave is characteristically similar to the harmonic under the tortoise coordinate x, while the wave piles up near the two horizons and the wavelength tends to its maximum as the potential approaches to the peak under the radial coordinate r.

Key words: scalar field, PÖshl–Teller potential, near extremal SdS space