Abstract
The Sasaki-Nakamura transformation gives a short-ranged potential and a convergent source term for the master equation of perturbations in the Kerr space-time. In this paper, we study the asymptotic behavior of the transformation, and present a new relaxed necessary and sufficient condition of the transformation to obtain the short-ranged potential in the assumption that the transformation converges in the far distance. Also, we discuss quasinormal mode frequencies which are determined by the information around the peak of the potential in the WKB analysis. Finally, in the extreme Kerr limit, $a/M \to 1$, where $M$ and $a$ denote the mass and spin parameter of a Kerr black hole, respectively, we find the peak location of the potential, $r_p/M \lesssim 1 + 1.8 \,(1-a/M)^{1/2}$ by using the new transformation. The uncertainty of the location is as large as that expected from the equivalence principle.
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