Heun Functions and Some of Their Applications in Physics

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.


Introduction
Most of the theoretical physics known today is described by using a small number of differential equations. If we study only linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe this problem. These equations have power series solutions with simple relations between consecutive coefficients and can be generally represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equation. Let us review some well known facts about linear second order differential equations. Differential equations are classified according to their singularity structure [1,2]. If a differential equation has no singularities over the full complex plane, it can only be a constant. Singularities are classified as regular or irregular singular points. If the coefficient of the first derivative has at most single poles, and the coefficient of the term without a derivative has at most double poles when the coefficient of the second derivative is unity, this second order differential equation has regular singularities. Then we have one regular solution while expanding around these singular points. The second solution around a regular singular point has a branch cut. If the poles of these coefficients are higher, we have irregular singularities and the general solution has essential singularities around these points [3]. As stated in Morse and Feshbach [4], an example of a second order differential equation with one regular singular point is This equation has one solution which is constant. The second solution blows up at infinity. The differential equation has one irregular singularity at infinity which gives an essential singularity at this point. The equation has two regular singular points, at zero and at infinity. In physics an often used equation is the hypergeometric equation This equation has three regular singular points, at zero, one and infinity. Jacobi, Legendre, Gegenbauer, Tchebycheff equations are special forms of this equation. When the singular points at z=1 and z equals infinity are " coalesced" [5] at infinity, we get the confluent hypergeometric equation with an essential singularity at infinity and a regular singulariy at zero. Bessel, Laguerre, Hermite equations can be reduced to this form. An important property of all these equations is that they allow infinite series solutions about one of their regular singular points and a recursion relation can be found between two consecutive coefficients of the series. This fact allows us having an idea about the general properties of the solution, as asymptotic behaviour at distant points, the radius of convergence of the series, etc.
A new equation was introduced in 1889 by Karl M. W. L. Heun [6]. This is an equation with four regular singular points. This equation is discussed in the book edited by Ronveaux [7]. All the general information we give below is taken from this book. As discussed there [8], any equation with four regular singular points can be transformed to the equation given below: There is a relation between the constants given as a + b + 1 = c + d + e . Then this equation has regular singularities at zero, one, f and infinity. If we try to obtain a solution in terms of a power series, one can not get a recursion relation between two consecutive coefficients. Such a relation exists between at least three coefficients. A simple solution as an integral transform also can not be found [1].
One can obtain different confluent forms of this equation. When we "coalesce" two regular singular points at zero and infinity, we get the confluent Heun equation [9]: Special forms of this equation are obtained in problems with two Coulombic centers [10], whose special form is the spheroidal equation [10], Another form is the algebraic form of the Mathieu equation [10]: If we coalesce two regular singular points pairwise, we obtain the double confluent form [11]: Here D = z d dz . If we equate the coefficients α i , i = −1, 1 and set B i , i = −1, 1 to zero, we can reduce the new equation to the Mathieu equation [12], an equation with two irregular singularities at zero and at infinity. Another form is the biconfluent form, where three regular singularities are coalesced. The result is an equation with a regular singularity at zero and an irregular singularity at infinity of higher order [13]: The anharmonic equation in three dimensions can be reduced to this equation [14]: In the triconfluent case, all regular singular points are "coalesced" at infinity which gives the equation below [15]: These different forms are used in different physics problems.
In SCI I found 94 papers when I searched for Heun functions. More than two thirds of these papers were published in the last ten years. The rest of the papers were published between 1990 and 1999, except a single paper in 1986 on a mathematical problem [16] . This shows that although the Heun equation was found in 1889, it was largely neglected in the physics literature until recently. Earlier papers on this topic are mostly articles in mathematics journals. The list of books on this topic also is not very long. There is a book edited by A.Ronveau, which is a collection of papers presented in the "Centennial Workshop on Heun's Equations: Theory and Application. Sept McLachlan [18] . These are on functions which are special cases of the Heun equation. Some well known papers on different mathematical properties of these functions can be found in references [19] - [23]. A reason why more physicists are interested in the Heun equation recently may be, perhaps, a demonstration of the fact that we do not have simple problems especially in " the theoretical theory" section of particle physics anymore and people doing research on this field have to tackle more difficult problems, either with more difficult metrics or in higher dimensions. Both of these extensions may necessitate the use of the Heun functions among the solutions. We can give the Eguchi-Hanson case as an example. The wave equation in the background of the Eguchi-Hanson metric [24] in four dimensions has hypergeometric functions as solutions [25] whereas the Nutku helicoid [26,27] metric, another instanton metric which is little bit more complicated than the Eguchi-Hanson one, gives us Mathieu function solutions [28], a member of the Heun function set. We also find that the same equation in the background of the Eguchi-Hanson metric, trivially extended to five dimensions, gives Heun type solutions [29].
Note that the problem need not be very complicated to end up with these equations. We encounter Mathieu functions if we consider two dimensional problems with elliptical boundaries [30]. Let us use x = 1 2 a cosh µ cos θ, y = 1 2 a sinh µ sin θ, where a is the distance from the origin to the focal point. Then the Helmholtz equation can be written as which separates into two equations a, k, h are constants given in [30]. The solutions to these two equations can be represented as Mathieu and modified Mathieu functions. If we combine different inverse powers of r, starting from first up to the fourth, or if we combine the quadratic potentials with inverse even powers of two, four and six, we see that the solution of the Schrodinger equation involves Heun functions [31]. Solution of the Schrödinger equation to symmetric double Morse potentials, like V (x) = B 2 4 sinh2x − (s + 1 2 )Bcoshx , where s = (0, 1/2, 1, ) [32], also needs these functions. Similar problems are treated in references [33,34,35].
o In atomic physics further problems such as separated double wells, Stark effect, hydrogen molecule ion use these functions. Physics problems which end up in these equations are given in the book by S.Y. Slavyanov and S. Lay [36].
Here we see that even the Stark effect, hydrogen atom in the presence of an external electric field, gives rise to this equation. As described in page 166 of Slavyanov's book, cited above (original reference is Epstein [37], also treated by S. Yu Slavyanov [38]), when all the relevant constants, namely Planck constant over 2π, electron mass and electron charge are set to unity, the Schrodinger equation for the hydrogen atom in a constant electric field of magnitude F in the z direction is given by Here ∆ is the laplacian operator.
Using parabolic coordinates, where the cartesian ones are given in terms of the new coordinates by x = √ ξηcosφ, y = √ ξηsinφ, z = ξ−η 2 and writing the wave function in the product form we get two separated equations: Here β 1 and β 2 are separation constants that must add to one. We note that these equations are of the biconfluent Heun form.
The hydrogen molecule also is treated in reference [36], (original reference is [39]). When the hydrogen-molecule ion is studied in the Born-Oppenheimer approximation, where the ratio of the electron mass to the proton mass is very small, one gets two singly confluent Heun equations if the prolate spheroidal coordinates ξ = r1+r2 2c , η = r1+r2 2c are used. Here c is the distance between the two centers. Assuming we achieve separation into two equations: Both equations are of the confluent Heun type. If we mention some recent papers on atomic physics with Heun type solutions we find three relatively recent papers which treat atoms in magnetic fields: o Exact low-lying states of two interacting equally charged particles in a magnetic field are studied in by Truong and Bazzali [40].
o The energy spectrum of a charged particle on a sphere under a magnetic field and Coulomb force is studied by Ralko and Truong [41]. Both papers reduce the problem to a biconfluent Heun equation. o B.S. Kandemir presented an analytical analysis of the two-dimensional Schrödinger equation for two interacting electrons subjected to a homogeneous magnetic field and confined by a two-dimensional external parabolic potential.
Here also a biconfluent Heun equation is used [42].
o Dislocation movement in crystalline materials, quantum diffusion of kinks along dislocations are some solid state applications of this equation. The book by S.Y. Slavyanov and S. Lay [36] is a general reference on problems solved before 2000.
o In a relatively recent work P. Dorey, J. Suzuki, R. Tateo [43] show that equations in finite lattice systems also reduce to Heun equations.
In the rest of this paper I will comment only on papers on particle physics and general relativity.
o In a relatively early work, Teukolsky studied the perturbations of the Kerr metric and found out that they were described by two coupled singly confluent Heun equations [44].
o Quasi-normal modes of rotational gravitational singularities were also studied by by E.W. Leaver [45] with Heun type equations.  [46]. D. Batic and H. Schmid also studied the Dirac equation for the Kerr-Newman metric and computed its propagator [47]. They found that the equation satisfied is a form of a generalized Heun equation described in Reference [46]. In later work Batic, with collaborators, continued studying Heun equations and their generalizations [48,49].
Prof. P.P. Fiziev studied problems whose solutions are Heun equations extensively: o In a paper published in gr-qc/0603003, he studied the exact solutions of the Regge-Wheeler equation in the Schwarschild black hole interior [50]. In another paper he obtained exact solutions of the same equation and quasi-normal modes of compact objects [51]. o Mirjam Cvetic, and Finn Larsen studied grey body factors and event horizons for rotating black holes with two rotation parameters and five charges in five dimensions. When the Klein Gordon equation for a scalar particle in this background is written, one gets a confluent Heun equation. In the asymptotic region this equation turns into the hypergeometric form [61]. When they studied the similar problem for the rotating black hole with four U (1) charges, they again obtained a confluent Heun equation for the radial component of the Klein Gordon equation, which they reduced to the hypergeometric form by making approximations [62]. These two papers are partly repeated in [63]. Same equations were obtained which were reduced to approximate forms which gave solutions in the hypergeometric form. Other relevant references I could find are listed as references [64] - [71].
I first "encountered" this type of equation when we tried to solve the scalar wave equation in the background of the Nutku helicoid instanton [26]. In this case one gets the Mathieu equation which is a special case of the Heun class [28].
o The helicoid instanton is a double-centered solution. As remarked above, for the simpler instanton solution of Eguchi-Hanson [24], hypergeometric solutions are sufficient [25]. Here one must remark that another paper using the Eguchi-Hanson metric ends up with the confluent Heun equation [72]. These two papers show that sometimes judicious choice of the coordinate system and separation ansatz matters. o Sucu andÜnal obtained closed solutions for the spinor particle written in the background of the Nutku helicoid instanton metric [25]. One can show that these solutions can be expanded in terms of Mathieu functions if one attempts to use the separation of variables method, as described by L.Chaos-Cador and E. Ley-Koo [73]. In the subsequent sections I will summarize some work Tolga Birkandan and I have done on this topic [29,74,75].
2 Dirac equation in the background of the Nutku helicoid metric [74,75] The Nutku helicoid metric is given as where 0 < r < ∞, 0 ≤ θ ≤ 2π, y and z are along the Killing directions and will be taken to be periodic coordinates on a 2-torus [28]. If we make the following transformation the metric is written as − sin 2θdydz + (sinh 2 x + cos 2 θ)dz 2 ]. We write the system in the form Lψ = Λψ, where L is the Dirac operator, Λ is the eigenvalue, and try to obtain the solutions for the different components. We use the Newman-Penrose formalism [76,77] to write the Dirac equation, with four components, in this metric. We use a solution of the type ψ i = e i(kyy+kzz) Ψ i (x, θ). We also use k y = kcosφ, k z = ksinφ. The transformation Ψ 1,2 = 1 √ sinh2x f 1,2 is used for the upper components to have similar equations for all components. Then these equations read: We solve our equations in terms of f 1,2 and substitute these expressions in equations, given above. This substitution gives us second order, but uncoupled equations for the lower components: We can separate this equation into two ordinary differential equations by the ansatz Ψ 3,4 = R(x)S(θ − φ). Using this ansatz gives us two ordinary differential equations. The equation for S reads where Θ = (θ − φ). This equation is of the Mathieu type and the solution can be written immediately.
where Se, So are the even and odd Mathieu functions. The solutions should be periodic in the angular variable Θ. This fact forces n, the separation constant, to take discrete values. It is known that the angular Mathieu functions satisfy an orthogonality relation such that functions with different n values are perpendicular to each other. Here, we integrate the angular variable from zero to 2π. The equation for R(x) reads This solution is of the double confluent form which can be reduced to the form by several transformations. Here C 1 , C 2 , D 1 , D 2 are arbitrary constants. A 6 , b are given in the original reference [74]. As a result of this analysis we see that the solutions of the Dirac equation, written in the background of the Nutku helicoid metric, can be expressed in terms of Mathieu functions, which is a special form of Heun function [74,75]. Mathieu function is a related but much more studied function with singularity structure sme as the double confluent Heun equation. o One can also show [74,75] that one can use the similar metric in five dimensions and obtain, in general, double confluent Heun functions which can be reduced to Mathieu functions.

Scalar field in the background of the extended
Eguchi-Hanson solution [29] To go to five dimensions, we can add a time component to the Eguchi-Hanson metric [24] so that we have where This is a vacuum solution.
If the variable transformation r = a √ cosh x is made, the solution can be expressed as We tried to express the equation for the radial part in terms of u = a 2 +r 2 2a 2 to see the singularity structure more clearly. Then the radial differential operator reads This operator has two regular singularities at zero and one, and an irregular singularity at infinity, the singularity structure of the confluent Heun equation. This is different from the hypergeometric equation, which has regular singularities at zero, one and infinity. The angular solution is in terms of hypergeometric functions. [(2 cos (θ) + 2)