Scattering state and bound state of scalar field in Schwarzschild   spacetime: Exact solution
  The main aim of this paper is twofold. (1) Exact solutions of a scalar field in the Schwarzschild spacetime are presented. The exact wave functions of scattering states and bound-states are presented. Besides the exact solution, we also provide explicit approximate expressions for bound-state eigenvalues and scattering phase shifts. (2) By virtue of the exact solutions, we give a direct calculation for the discontinuous jump on the horizon for massive scalar fields, while in literature such a jump is obtained from an asymptotic solution by an analytic extension treatment. 
Wen-Du Li, Yu-Zhu Chen, and Wu-Sheng Dai
gr-qc hep-th