High-overtone normal modes of Schwarzschild black holes

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Abstract
The frequencies of normal-mode oscillations of Schwarzschild black holes are studied using a higher-order WKB approach. The method is mathematically equivalent to locating the poles in the transmission amplitude for one-dimensional quantum mechanical tunnelling through a potential barrier. The transmission amplitude is expressed in terms of contour integrals of WKB functions around turning points of the effective barrier in the Regge-Wheeler equation. In the limit of large overtone number n, the dimensionless frequency sigma identical to omega M (M=black-hole mass) for l=2 gravitational modes is given by sigma =-i(2n+1)/8+1/2A(1+i)(2n+1)-12/+. . ., where A approximately=1.0. The compare their results with numerical computations of Leaver and find striking and unexplained differences.