Solutions for confluent and double-confluent Heun equations

El-Jaick, Léa Jaccoud; Figueiredo, Bartolomeu D. B.

doi:10.1063/1.2970150

Cited by 51 publications

(78 citation statements)

References 29 publications

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“…Solution to symmetric double Morse potentials also needs these functions, like V (x) = B 2 /4sinh2x − (s + 1/2)Bcoshx where s = (0, 1/2, 1, ...) [33]. Similar problems are treated in references [34], [35] and [36] o In atomic physics further problems such as separated double wells, Stark effect, hydrogen molecule ion use these functions. Physics problems which end up with these equations are given in the book by S.Y.…”

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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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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“…A second subgroup, with different regions of convergence, results from the transformations of the DCHE and cannot be obtained as limits of the expansions in series of Coulomb wave functions. In effect, if U(z) = U(B 1 , B 2 , B 3 ; ω, η; z) denotes one solution for the DCHE, new solutions can be generated by using the transformations [3] …”

Section: Solutions For the Double-confluent Heun Equationmentioning

confidence: 99%

“…Later on [2,3,4] we have found that there are two other physically relevant equations whose solutions can also be derived from the Leaver solutions for the CHE and DCHE by means of a procedure called Whittaker-Ince limit. Further, from solutions of the CHE and/or DCHE, we can find solutions for the Mathieu, Whittaker-Hill and spheroidal equations [1,5].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the CHE and the DCHE admit a (Whittaker-Ince) limit which changes the nature of the singularity at z = ∞, keeping unaltered the other singular points. This limit is given by [2,3] ω → 0, η → ∞, such that 2ηω = −q, (Whittaker-Ince limit) (3) where q is a nonvanishing constant. The Whittaker-Ince limit of the CHE is…”

Section: Introductionmentioning

confidence: 99%

“…For QES problems obeying equations of the Heun family [3], that part of the spectrum may be derived from finite-series solutions if these are known. Indeed, a problem is also said to be QES if it admits solutions given by finite series whose coefficients necessarily satisfy three-term or higher order recurrence relations [13], and is said to be exactly solvable if its solutions can be expressed by hypergeometric functions.…”

Section: Introductionmentioning

confidence: 99%

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Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions

El-Jaick¹,

Figueiredo²

2013

J. Phys. A: Math. Theor.

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The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index n runs from minus to plus infinity [E. W. Leaver, J. Math. Phys. 27, 1238Phys. 27, (1986]. First we show that, in contrast to the D'Alembert test, under certain conditions the Raabe test assures that the domains of convergence of these solutions include an additional singular point. Further, by using a limit proposed by Leaver, we obtain solutions for the double-confluent Heun equation (DCHE). In addition, we get solutions for the so-called Whittaker-Ince limit of the CHE and DCHE. For these four equations, new solutions are generated by transformations of variables. In the second place, for each of the above equations we obtain one-sided series solutions (n ≥ 0) by truncating on the left the two-sided series. Finally we discuss the time dependence of the Klein-Gordon equation in two cosmological models and the spatial dependence of the Schrödinger equation to a family of quasi-exactly solvable potentials. For a subfamily of these potentials we obtain infinite-series solutions which converge and are bounded for all values of the independent variable, in opposition to a common belief.

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Solving second‐order differential equations in terms of confluent Heun's functions

Aldossari

2024

Math Methods in App Sciences

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Solutions for confluent and double-confluent Heun equations

Cited by 51 publications

References 29 publications

Heun Functions and Their Uses in Physics

Heun Functions and Their Uses in Physics

Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions

Solving second‐order differential equations in terms of confluent Heun's functions

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