Analysis of the three possible event horizons of anti-de Sitter space-time in the Klein-Gordon equation

In this paper, we calculate exactly the solution of Klein-Gordon equation in anti-de Sitter space-time. Due to spherical symmetry, the angular wave functions are given by spherical harmonics. The radial wave functions are given by local Heun functions. We analyze the behavior of the radial wave function in regions near three possible event horizon positions (r0,r1,r2) and investigate the decay rate, Hawking radiation and Hawking temperature for these three cases. Comparing the results, we verify that if the decay rate and Hawking radiation are greater, then the closer the event horizon is to the singularity of the black hole. Finally, we analyze the special case of a small black hole limit.