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Register a new userElementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
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Added by
Pierre-Louis Giscard
Publication date
2020
fields
Physics
and research's language is
English
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Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide explicit integral representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly-confluent and triconfluent, with integrals involving only rational functions and exponential integrands. All the series are illustrated with concrete examples of use. These results stem from the technique of path-sums, which we use to evaluate the path-ordered exponential of a variable matrix chosen specifically to yield Heun functions. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.
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Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlev{e} equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lam{e} and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two-way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable. The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2008. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given.
We find confluent Heun solutions to the radial equations of two Halilsoy-Badawi metrics. For the first metric, we studied the radial part of the massless Dirac equation and for the second case, we studied the radial part of the massless Klein-Gordon equation.
We provide a set of diagonals of simple rational functions of three and four variables that are squares of Heun functions. These Heun functions obtained through creative telescoping, turn out to be either pullbacked $_2F_1$ hypergeometric functions and in fact classical modular forms. We also obtain Heun functions that are Shimura curves as solutions of telescopers of rational functions.
Quantum theory on manifolds with boundaries have been studied extensively through von Neumann analysis of self adjoint operators. We approach the issues through introduction of singular $delta$ and $delta$ potentials. The advantages of this are pointed out as a model for black hole and in several other examples.
Classical Virasoro conformal blocks are believed to be directly related to accessory parameters of Floquet type in the Heun equation and some of its conflue
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