Second order perturbative calculation of quasinormal modes of Schwarzschild black holes

We analytically calculate to second order the correction to the asymptotic form of quasinormal frequencies of four dimensional Schwarzschild black holes based on the monodromy analysis proposed by Motl and Neitzke. According to our calculation, all the second order corrections are purely imaginary. For gravitational axial perturbation, the numerical values of the imaginary part of the correction are in good agreement with the available numerical results. However, our result for the real part shows sizeable discrepancy with the numerical one for large angular momentum ($l=6$). This might indicate certain limitations in the numerical methods used in the literature and more study is needed to settle this issue.