Spectral Problems for Quasinormal Modes of Black Holes

Abstract

This is an unconventional review article on spectral problems in black hole perturbation theory. Our purpose is to explain how to apply various known techniques in quantum mechanics to such spectral problems. The article includes analytical/numerical treatments, semiclassical perturbation theory, the (uniform) WKB method and useful mathematical tools: Borel summations, Padé approximants, and so forth. The article is not comprehensive, but rather looks into a few examples from various points of view. The techniques in this article are widely applicable to many other examples.

Semi-analytic expressions for quasinormal modes of slowly rotating Kerr black holes

“…where we used r ± = M ± √ M 2 − a 2 and the numerical data of e − j and e − jk [1,2,17]. We confirmed that the Sasaki-Nakamura equation reproduces the same result in the end.…”

“…where we used r ± = M ± √ M 2 − a 2 and the numerical data of e − j [1,17,21]. We note that we can obtain the same result from the Sasaki-Nakamura equation if we use the recursion relation for e − j [3] (see Appendix B).…”

“…We should note that the coefficients e ± j and e ± jk do not depend on the small parameters α ± j , and thus our task is to know them. Their numerical data are found in [1,2,17]. 3…”

“…3 In [17], we recalculated the coefficients e ± j and e ± jk . We used the recursion relations for e − j [3] and their higher order extensions to evaluate the numerical error.…”

An alternative to the Teukolsky equation

“…We compute slow spin corrections to the resonance frequency. To do so, we use a systematic formalism proposed in [22]. The potential (2) has a natural deformation parameter.…”