Eigenvalue for angular Teukolsky equation via accessory parameter for Painlevé V

Abstract

The purpose of this dissertation is to present an alternative way to compute the eigenvalues for spheroidal harmonics, in view of its applications to arbitrary spin quasi-normal frequencies of Kerr black hole. The alternative is based on the relation between the connection problem of the angular Teukolsky Master Equation (TME) and the dependence of the Painlevé V transcendent on monodromy data. The latter has an expansion in terms of irregular conformal blocks, uncovered by the AGT correspondence, which can in principle be used for explicit calculations. The isomonodromic deformations in the angular TME is translated to two conditions on the Painlevé V transcendent which are solved to find the expansion of the accessory parameter of the angular TME and consequently the first terms of the expansion of the eigenvalue ₛλₗₘ.