Scattering from an AdS_3 bubble and an exact AdS_3

We investigate the close relationship between the potential and absorption cross section for test fields in an AdS${}_3$ bubble(a 5D black hole) and an exact AdS${}_3$. There are two solutions in type IIB string theory: an AdS${}_3$ bubble corresponds to the dilatonic solution, while an exact AdS${}_3$ is the non-dilatonic solution. In order to obtain the cross section for an AdS${}_3$ bubble, we introduce the {out}-state scattering picture with the AdS${}_3$-AFS matching procedure. For an exact AdS${}_3$, one considers the {in}-state scattering picture with the AdS${}_3$-AdS${}_3$ matching. Here the non-normalizable modes are crucially taken into account for the matching procedure. It turns out that the cross sections for the test fields in an AdS${}_3$ bubble take the same forms as those in an exact AdS${}_3$. This suggests that in the dilute gas and the low energy limits, the S-matrix for an AdS${}_3$ bubble can be derived from an exact AdS${}_3$ space.