Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system

Choun, Yoon Seok

doi:10.48550/arxiv.1302.7309

Cited by 6 publications

(16 citation statements)

References 29 publications

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“…There are four kinds of confluent forms of Heun equation [41,39,32,50,7] such as the Confluent Heun [6,29,35], Doubly-Confluent Heun [20], Biconfluent Heun [8] and Triconfluent Heun equations [26]. We can derive these four confluent forms from Heun equation by combining two or more regular singularities to each other to take form an irregular singularity.…”

Section: Discussionmentioning

confidence: 99%

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Asymptotic behavior of Heun function and its integral formalism

Choun

2013

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The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric 2 F 1 , 1 F 1 and 0 F 1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.In this paper, applying three term recurrence formula [9], we consider asymptotic behaviors of Heun function and its integral formalism including all higher terms of A n 's. 1 We show how the power series expansion of Heun functions can be converted to closed-form integrals for all cases of infinite series and polynomial. One interesting observation resulting from the calculations is the fact that a 2 F 1 function recurs in each of sub-integral forms: the first sub-integral form contains zero term of A n s, the second one contains one term of A n 's, the third one contains two terms of A n 's, etc.Applying three term recurrence formula, we consider asymptotic behaviors of Heun functions and their radius of convergences. And we show why Poincaré-Perron theorem is not always applicable to the Heun equation.In the appendix, I apply the power series expansion and my integral formalism of Heun function to "The 192 solutions of the Heun equation" [34]. Due to space restriction final equations for all 192 Heun functions is not included in the paper, but feel free to contact me for the final solutions. Section 6 contains two additional examples using integral forms of Huen function.This paper is 4th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. The previous paper in series deals with the power series expansion in closed forms of Heun function. The next paper in the series describes the power series expansion of Mathieu function and its integral formalism analytically.

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Section: Discussionmentioning

confidence: 99%

“…Since we substitute (3.32a), (3.33), (3.35), (3.36) and including all integral forms of y m (x) terms where m ≥ 4 into (3.30), we obtain (3.31). 8 Put c 0 = 1 as λ=0 for the first kind of independent solutions of Heun equation and c 0 = a −1 (1 + a)…”

Section: Infinite Seriesmentioning

confidence: 99%

Asymptotic behavior of Heun function and its integral formalism

Choun

2013

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“…1. "Approximative solution of the spin free Hamiltonian involving only scalar potential for the q − q system" [15] -In order to solve the spin-free Hamiltonian with light quark masses we are led to develop a totally new kind of special function theory in mathematics that generalize all existing theories of confluent hypergeometric types. We call it the Grand Confluent Hypergeometric Function.…”

Section: Series "Special Functions and Three Term Recurrence Formula ...mentioning

confidence: 99%

The generating functions of Lame equation in Weierstrass's form

Choun

2013

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Lame equation arises from deriving Laplace equation in ellipsoidal coordinates; in other words, it's called ellipsoidal harmonic equation. Lame function is applicable to diverse areas such as boundary value problems in ellipsoidal geometry, chaotic Hamiltonian systems, the theory of Bose-Einstein condensates, etc.By applying generating function into modern physics (quantum mechanics, thermodynamics, black hole, supersymmetry, special functions, etc), we are able to obtain the recursion relation, a normalization constant for the wave function and expectation values of any physical quantities. For the case of hydrogen-like atoms, generating function of associated Laguerre polynomial has been used in order to derive expectation values of position and momentum. By applying integral forms of Lame polynomial in Weierstrass's form in which makes B n term terminated[21], I consider generating function of it including all higher terms of A n 's. 1 This paper is 8th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 4 for all the papers in the series. Previous paper in series deals with the power series expansion and the integral formalism of Lame equation in Weierstrass's form and its asymptotic behavior [21]. The next paper in the series describes analytic solution for grand confluent hypergeometric function [23].

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“…1. "Approximative solution of the spin free Hamiltonian involving only scalar potential for the q − q system" [22] -In order to solve the spin-free Hamiltonian with light quark masses we are led to develop a totally new kind of special function theory in mathematics that generalize all existing theories of confluent hypergeometric types. We call it the Grand Confluent Hypergeometric Function.…”

Section: Applicationmentioning

confidence: 99%

The power series expansion of Mathieu function and its integral formalism

Choun

2013

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Mathieu ordinary differential equation is of Fuchsian types with the two regular and one irregular singularities. In contrast, Heun equation of Fuchsian types has the four regular singularities. Heun equation has the four kind of confluent forms: (1) Confluent Heun (two regular and one irregular singularities), (2) Doubly confluent Heun (two irregular singularities), (3) Biconfluent Heun (one regular and one irregular singularities), (4) Triconfluent Heun equations (one irregular singularity). For DLFM version [21], Mathieu equation in algebraic forms is also derived from the Confluent Heun equation by changing all coefficients δ = γ = 1 2 , ǫ = 0, α = q and q = λ+2q 4 . In this paper I apply three term recurrence formula [23] to the power series expansion in closed forms of Mathieu equation for infinite series and its integral forms including all higher terms of A n 's. One interesting observation resulting from the calculations is the fact that a modified Bessel function recurs in each of sub-integral forms: the first sub-integral form contains zero term of A ′ n s, the second one contains one term of A n 's, the third one contains two terms of A n 's, etc. Section 5 contains two additional examples of Mathieu function.This paper is 5th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. Previous paper in series deals with asymptotic behavior of Heun function and its integral formalism [25]. The next paper in the series describes the power series expansion in closed forms of Lame equation in the algebraic form and its integral forms [27].

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Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system

Cited by 6 publications

References 29 publications

Asymptotic behavior of Heun function and its integral formalism

Asymptotic behavior of Heun function and its integral formalism

The generating functions of Lame equation in Weierstrass's form

The power series expansion of Mathieu function and its integral formalism

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