The generalized Heun equation in QFT in curved spacetimes

Batic, Davide; Schmid, Harald; Winklmeier, Monika

doi:10.1088/0305-4470/39/40/019

Cited by 25 publications

(36 citation statements)

References 11 publications

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“…In view of the results obtained in [5] since the Kerr-Newman-deSitter metric becomes the Kerr-Newman metric in the limit of a vanishing cosmolog-ical constant it follows that the generalized Heun equation is a confluent case of the HHE for Λ → 0. Finally, since it has been proved in [12] that the neutrino equation can be separated in the class 0 of algebraically special Petrov type-D vacuum solutions with cosmological constant, whereas the massive Dirac equation can be separated only in the Carter's subclass A, our result raises the question whether the separated neutrino and massive Dirac equations can also be reduced to a HHE in such more general metrics.…”

Section: Discussionmentioning

confidence: 89%

“…In the case Λ = = 0 the components S ± of the angular eigenfunctions are spinweighted spherical harmonics [9,10] whereas for Λ = 0 and = 0 the radial and angular eigenfunctions satisfy a generalized Heun equation [5,11]. In the next section we show that in the more general case Λ = 0 = the functions R ± and S ± satisfy the hypergeneralized Heun Eq.…”

Section: The Dirac Equation In the Kerr-newman-desitter Metricmentioning

confidence: 82%

“…Moreover, for α = 0 = β 2 the GHE becomes the HE and the point at infinity is now a simple singularity. The GHE arises in physics from the sepa-rated radial and angular parts of a massive Dirac equation in the presence of a Kerr-Newman black hole and in the context of static perturbations for the non-extremal Reissner-Nordström metric [5]. At this point, one might ask if there is some physical context in which the following generalization of the previous equations…”

Section: Introductionmentioning

confidence: 99%

“…To underline the importance of Eq. (1) we recall that it represents a generalization of the Fuchsian differential equation with five simple singularities studied by Schmidt in [6] and it contains, as a confluent special case, the generalized Heun equation [4,5] and thus the Heun equation as well. The aim of our work is to provide the first example where the HHE arises in physics.…”

Section: Introductionmentioning

confidence: 99%

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The hypergeneralized Heun equation in quantum field theory in curved space-times

Batic

Sandoval

2010

Open Physics

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Section: Discussionmentioning

confidence: 89%

Section: The Dirac Equation In the Kerr-newman-desitter Metricmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The hypergeneralized Heun equation in quantum field theory in curved space-times

Batic

Sandoval

2010

Open Physics

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“…We must note that even the simplest blackhole metric, the Schwarzschild, has solutions in the [63]. They found that the equation satisfied is a form of a general Heun equation described in Reference [62]. In later work Batic, with collaborators continued studying Heun equations and their generalizations [64].…”

Section: Some Examples Of the Heun Equation In Physical Applicationsmentioning

confidence: 99%

Heun Functions and Their Uses in Physics

Hortaçsu

2012

Mathematical Physics

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Most of the theoretical physics known today is described 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 Painlevé 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, Lamé 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 2010. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. An integral transform solution using simpler functions also is not obtainable. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given. * This paper is a revised and few times updated version of the conference talk by the same author, published in

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A conditionally exactly solvable generalization of the inverse square root potential

Ishkhanyan

2016

Physics Letters A

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We present a conditionally exactly solvable singular potential for the one-dimensional Schrödinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equation for the energy spectrum which is written using two Hermite functions of non-integer order. In specific auxiliary variables this equation becomes a mathematical equation that does not refer to a specific physical context discussed. In the two-dimensional space of these auxiliary variables the roots of this equation draw a countable infinite set of open curves with hyperbolic asymptotes. We present an analytic description of these curves by a transcendental algebraic equation for the involved variables. The intersections of the curves thus constructed with a certain cubic curve provide a highly accurate description of the energy spectrum.

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The generalized Heun equation in QFT in curved spacetimes

Cited by 25 publications

References 11 publications

The hypergeneralized Heun equation in quantum field theory in curved space-times

The hypergeneralized Heun equation in quantum field theory in curved space-times

Heun Functions and Their Uses in Physics

A conditionally exactly solvable generalization of the inverse square root potential

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