The zeroth-order solution satisfies the homogeneous spherical Bessel equation, and must be a linear combination of the spherical Bessel functions of the first and second kinds, and . Here, we demand the compatibility with the horizon solution (80). Since and , does not match with the horizon solution at the leading order of . Therefore, we have
The procedure to obtain was described in detail in . Using the Green function , Equation (84) may be put into an indefinite integral form, or Appendix D of . Using those formulae, for we have
Here, we again perform the matching with the horizon solution (80). It should be noted that , given by Equation (87), is regular in the limit except for the term . By examining the asymptotic behavior of Equation (87) at , we find , i.e., the solution is regular at . As for , it only contributes to the renormalization of . Hence, we set and the transmission amplitude is determined to as
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