Quantum field theory applications of heun type functions

Birkandan, T.; Hortaçsu, M.

doi:10.1016/s0034-4877(17)30022-8

Cited by 21 publications

(15 citation statements)

References 28 publications

(48 reference statements)

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“…The general Heun equation and its confluent forms appear in many fields of modern physics, such as general relativity, astrophysics, hydrodynamics, atomic and particle physics, etc. (see, e.g., [3,6,18,22,25,2,7]). A vast list of references to numerous physical applications, especially in general relativity, can be found in [9].…”

Section: Introductionmentioning

confidence: 99%

On evaluation of the confluent Heun functions

Motygin

2018

2018 Days on Diffraction (DD)

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In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points -two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given. arXiv:1804.01007v1 [math.NA] 2 Apr 2018 2 Statement and basic notationsWe use the following form of the confluent Heun equation:This second order linear differential equation has regular singularities at z = 0 and 1, and an irregular singularity of rank 1 at z = ∞ (see e.g. [20]). The parameter q ∈ C is usually referred to as an accessory or auxiliary parameter and γ, δ, ε, α (also belonging to C) are exponent-related parameters.It is important to note that in this paper the parameters ε and α are assumed to be independent. Below, we will use notation c H (q, α, γ, δ, ε; z) or c H (z) for brevity. There are 6 local solutions of equation (1). The Frobenius method can be used to derive local power-series solutions to (1) near z = 0 and z = 1 (two per a singular point), while two solutions at z = ∞ can be obtained in the form of asymptotic series. In § 3 we will present the local solutions near the point z = 0. One of them is analytic in a vicinity of zero and if γ is not a nonpositive integer, we normalize this solution to unity at z = 0 and call it the local confluent Heun function. It is denoted by c Hl (q, α, γ, δ, ε; z). For the second Frobenius local solution, we will use the notation c Hs(q, α, γ, δ, ε; z). When γ is a nonpositive integer, one solution of (1) is analytic in a vicinity of z = 0 but it is equal to zero at z = 0, whereas the second solution can be normalized to unity at zero but generally it is not analytic. Following [16], the normalized solution will be denoted by c Hl (z) and another one by c Hs(z).

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Section: Introductionmentioning

confidence: 99%

On evaluation of the confluent Heun functions

Motygin

2018

2018 Days on Diffraction (DD)

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“…A long endeavor was necessary to label Teukolsky Master Equations as belonging to the Heun class [58]. Only recently the equation given by 't Hooft [140] was shown to belong to the Heun class if it were not modified [141]. When modified the solution is the manageable hypergeometric function.…”

Section: Resultsmentioning

confidence: 99%

Heun Functions and Some of Their Applications in Physics

Hortaçsu

2018

Advances in High Energy Physics

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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 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 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. * This paper is a revised and many times updated version of the conference talk by the same author, published in

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“…After a formal analysis of the radial part, one can see that the radial solution can be given in terms of confluent Heun functions [20]. General and confluent forms of the Heun function are encountered in many applications in physics, especially as solutions of the wave equations [21][22][23][24][25][26].…”

Section: In Our Example the Klein-gordon Equation Is Found Asmentioning

confidence: 99%

Symbolic and numerical analysis in general relativity with open source computer algebra systems

et al. 2018

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We study three computer algebra systems, namely SageMath (with SageManifolds package), Maxima (with ctensor package) and Python language (with GraviPy module), which allow tensor manipulation for general relativity calculations along with general algebraic calculations. We present a benchmark of these systems using simple examples. After the general analysis, we focus on the SageMath and SageManifolds system to derive, analyze and visualize the solutions of the massless Klein-Gordon equation and geodesic motion with Hamilton-Jacobi formalism. We compare our numerical result of the Klein-Gordon equation with the asymptotic form of the analytical solution to see that they agree. * Man = Manifold (4 , 'Man ' , r '\ mathcal { M } ') 4 5 # Define the parameter " M " ( mass ) : 6 M = var ( 'M ') 7 8 # Define the coordinates { t =0 , r =1 , theta (= th ) =2 , phi =3} with ranges 9 # ( BL = Boyer -Lidquist ) 10 BL . = Man . chart (r ' t r :(0 ,+ oo ) th :(0 , pi ) :\ theta ph :(0 ,2* pi ) :\ phi ') 11 12 # Define the metric " g " on manifold " Man ": 13 g = Man . lorentzian_metric ( 'g ') 14 15 # Enter the metric components : 16 g [0 ,0] = (1 -(2* M ) / r ) 17 g [1 ,1] = -1/(1 -(2* M ) / r ) 18 g [2 ,2] = -r^2 19 g [3 ,3] = -( r * sin ( th ) )^2 20 21 # Einstein tensor 40 ET = g . ricci () -(1/2) * g * g . ricci_scalar () 41 # Display all components 42 ET . set_name ( 'E ') 43 show ( ET . display () ) 44 # Display a single component 45 show ( ET [1 ,2]) ¦ ¥ The lines between 16-19 defines the Schwarzschild metric. In order to define the Kerr metric, we need to change this part with § ¤ 16 a = var ( 'a ') 17 rhosq = r^2+( a^2) * cos ( th )^2 18 Delta = r^2 -2* M * r + a^2 19 20 g [0 ,0] = (1 -(2* M * r ) / rhosq ) 21 g [1 ,1] = -rhosq / Delta 22 56 # We can analyze the Klein -Gordon equation 57 # to see how it is separated into 58 # radial and angular parts . 59 60 # Common factors will be divided 61 # This part should be given by the user 62 divideKGby = exp ( -I * omega * t ) * exp ( I * k * ph ) * sin ( th ) * R * S 63 finalKG = expand ( KG / divideKGby ) 64 65 # Extract the operands in the expression : 66 fkgops = finalKG . operands () 67 68 # Find radial and angular parts : 69 # lambda_aux is the separation constant 70 var ( ' lambda_aux ') 71 KGradialpart = lambda_aux 72 KGangularpart = -lambda_aux 73 for term in fkgops : 74 if diff ( term , r ) ==0: 75 KGangularpart = KGangularpart + term 76 else : 77 KGradialpart = KGradialpart + term 78 79 KGradialpart = expand ( KGradialpart * R) . simplify_full () . collect ( R ) . collect ( diff (R , r ) ) . collect ( diff (R ,r , r ) ) 80 KGangularpart = expand ( KGangularpart * S ) . simplify_full () . collect ( S ) . collect ( diff (S , th ) ) . collect ( diff (S , th , th ) ) 81 110 111 # Substitute numerical values for parameters 112 radial2 = radial2 . subs ( M =0.1 , omega =0.2 , k =2.0 , lambda_aux =2.0) 113 13 114 # Solve the system and plot the solution 115 radsol = desolve_system_rk4 ([ radial1 , radial2 ] ,[R , R_aux ] , ics =[0.3 ,1 ,0.5] , ivar ...

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Quantum field theory applications of heun type functions

Cited by 21 publications

References 28 publications

On evaluation of the confluent Heun functions

On evaluation of the confluent Heun functions

Heun Functions and Some of Their Applications in Physics

Symbolic and numerical analysis in general relativity with open source computer algebra systems

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