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# Antiquantization of the Double Confluent Heun Equation. The Teukolsky Equation

2019, Vol. 15, no. 1, pp.  79-85

Author(s): Salatich A. A., Slavyanov S. Y.

Different forms of the double confluent Heun equation are studied. A generalized Riemann scheme for these forms is given. An equivalent first-order system is introduced. This system can be regarded from the viewpoint of the monodromy property. A corresponding Painlevé equation is derived by means of the antiquantization procedure. It turns out to be a particular case of $P^3$. A general expression for any Painlevé equation is predicted. A particular case of the Teukolsky equation in the theory of black holes is examined. This case is related to the boundary between spherical and thyroidal geometries of a black hole. Difficulties for its antiquantization are shown.
Keywords: Double confluent Heun equation, antiquantization, Painlevé equation $P^3$, Teukolsky equation
Citation: Salatich A. A., Slavyanov S. Y., Antiquantization of the Double Confluent Heun Equation. The Teukolsky Equation, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp.  79-85
DOI:10.20537/nd190108

## References

[1] Slavyanov, S. Yu., “Painlevé Equations As Classical Analogues of Heun Equations”, J. Phys. A, 29:22 (1996), 7329–7335
[2] Slavyanov, S. Yu. and Lay, W., Special Functions: A Unified Theory Based on Singularities, Oxford Univ. Press, Oxford, 2000, 312 pp.
[3] Slavyanov, S. Yu. and Stesik, O. L., “Antiquantization of Deformed Heun-Class Equations”, Theoret. and Math. Phys., 186:1 (2016), 118–125          ; Teoret. Mat. Fiz., 186:1 (2016), 142–151 (Russian)
[4] Babich, M. and Slavyanov, S., “Antiquantization, Isomonodromy, and Integrability”, J. Math. Phys., 59:9 (2018), 091416, 11 pp.
[5] Staicova, D. and Fiziev, P., “The Spectrum of Electromagnetic Jets from Kerr Black Holes and Naked Singularities in the Teukolsky Perturbation Theory”, Trends in Particle Physics (Primorsko, Bulgaria, 2010)
[6] Kazakov, A. Ya. and Slavyanov, S. Yu., “Euler Integral Symmetries for the Confluent Heun Equation and Symmetries of the Painlevé Equation PV”, Theoret. and Math. Phys., 179:2 (2014), 543–549          ; Teoret. Mat. Fiz., 179:2 (2014), 189–195 (Russian)
[7] Slavyanov, S. Yu., “Kovalevskaya's Dynamics and Schrödinger Equations of Heun Class”, Operator Methods in Ordinary and Partial Differential Equations (Stockholm, 2000), Oper. Theory Adv. Appl., 132, eds. S. Albeverio, N. Elander, W. N. Everitt, P. Kurasov, Birkhäuser, Basel, 2002, 395–402
[8] Teukolsky, A. S., “Rotating Black Holes: Separable Wave Equations for Gravitational and Electromagnetic Perturbations”, Phys. Rev. Lett., 29:16 (1972), 1114–1118
[9] Staicova, D. and Fiziev, P., “The Spectrum of Electromagnetic Jets from Kerr Black Holes and Naked Singularities in the Teukolsky Perturbation Theory”, Astrophys. Space Sci., 332:2 (2011), 385–401
[10] Casals, M. and Micchi, L. F. L., Spectroscopy of Extremal (and Near-Extremal) Kerr Black Holes, 2019, arXiv: 1901.04586 [gr-qc]
[11] London, L. and Fauchon-Jones, E., On Modeling for Kerr Black Holes: Basis Learning, QNM Frequencies, and Spherical-Spheroidal Mixing Coefficients, 2019, arXiv: 1810.03550 [gr-qc]
[12] Lay, W., Bay, K., and Slavyanov, S. Yu., “Asymptotic and Numeric Study of Eigenvalues of the Double Confluent Heun Equation”, J. Phys. A, 31:42 (1998), 8521–8531
[13] Novaes, F. and de Cunha, B. C., Isomonodromy, Painlevé Transcendents and Scattering off of Black Holes, 2014, arXiv: 1404.5188 [hep-th]