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Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 612757, 8 pages http://dx.doi.org/10.1155/2015/612757

Research Article Exact Solution of the Curved Dirac Equation in Polar Coordinates: Master Function Approach

H. Panahi and L. Jahangiri

Department of Physics, University of Guilan, Rasht 41635-1914, Iran

Correspondence should be addressed to H. Panahi; [email protected]

Received 6 February 2015; Accepted 25 April 2015

Academic Editor: Ming Liu

Copyright © 2015 H. Panahi and L. Jahangiri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP 3 .

We show that the (2 + 1) curved Dirac equation in polar coordinates can be transformed into Schrodinger-like differential equation for upper spinor component. We compare this equation with the Schrodinger equation derived from shape invariance property of second order differential equations of mathematical physics. This formalism enables us to determine the electrostatic potential and relativistic energy in terms of master function and corresponding weight function. We also obtain the spinor wave function in terms of orthogonal polynomials.

1. Introduction

The Klein-Gordon and Dirac equations are relativistic wave equations which describe the behaviour of spin 0 and 1/2 particles, respectively. These equations were formulated in order to construct a unified theory involving quantum mechanics and special relativity. Undoubtedly, the Dirac equation is one of the remarkable achievements in modern physics because of its important role in particle physics by predicting the existence of antiparticles. In order to investigate the relativistic effects, this equation has been solved for different potentials like Coulomb [1], Rosen-Morse [2], Woods-Saxon [3], generalized Poschl-Teller [4], Yukawa [5], and so forth. The Dirac equation was first written in Minkowski spacetime but in order to study the behaviour of fermions in gravitational fields, this equation has been extended to curved space-time. The Dirac equation in curved space-time has been considered in different curved backgrounds due to its considerable applications in astrophysics, cosmology, and condensed matter [6–9]. On the other hand, in recent years, different methods have been applied in order to study the Dirac equation such as separation of variables [10, 11], supersymmetric quantum mechanics [12, 13], and NikiforovUvarov [14, 15]. As we know the supersymmetry [16] and shape invariance [17] formalism is one of the most important methods which has been successfully used in order to study exactly solvable quantum systems. In this formalism, two partner Hamiltonians can be factorized into product of two first order differential operators and the energy spectrum and wave functions of these two Hamiltonians are related to each other by supersymmetric relations [18]. On the other hand, it has been shown that the second order differential equations of mathematical physics and their associated differential equations have the properties of supersymmetry and shape invariance [19, 20]. By introducing the master function formalism, the authors could transform these equations into Schrodinger equation and obtained some well-known onedimensional shape invariant potentials. The master function formalism as an interesting method can be also used in order to study the relativistic and nonrelativistic quantum systems [21–23]. In [24], the curved Dirac equation in two- and three-dimensional spaces has been studied only in presence of magnetic field. In two-dimensional space, they let an imaginary value for radial component of electromagnetic potential and obtained different angular gauge fields in terms of master function. Now in this work we consider the curved Dirac equation in polar coordinates in the presence of electromagnetic field and show that after inserting a special relation between the components of electromagnetic potential we can obtain a Schrodinger-like second order differential equation Advances in High Energy Physics for the upper spinor component. We then study this equation from master function formalism.

Hence, the paper is organized as follows. In Section 2, we consider the Dirac equation in curved space-time in polar coordinates and construct a Schrodinger-like differential equation for the upper spinor component by using a unitary transformation. In Section 3, we review the factorization formalism of second order differential equations of mathematical physics obtained from the master function. Then, in Section 4, we show that the Schrodinger-like equation obtained from Dirac equation has the shape invariance symmetry in the framework of master function formalism.

where the spinorial affine connection Γ 𝜇 can be obtained as

Γ𝜇 = 1 8 𝜔𝜇 𝑐𝑑 [𝛾𝑐 , 𝛾𝑑 ] .

The spin connection 𝜔 𝜇 𝑐 𝑑 satisfies the following relation:

𝜔𝜇 𝑐 𝑑 = 𝑒𝑐 ]𝑒 𝜎 𝑑Γ ]𝜎 𝜇 + 𝑒𝑐 ]𝜕 𝜇 𝑒 ] 𝑑,

Therefore, the Dirac equation in curved space-time has the following form:

symmetry in the framework of master function formalism. The electrostatic potential, the energy spectrum, and the wave function of the Dirac equation are obtained in terms of master function and corresponding weight function. After that, we give an example and the other cases of electrostatic potential are given in the Appendix. At the end, we give a brief conclusion in Section 5. 2. The (2 + 1) Curved Dirac Equation in Polar Coordinates The Dirac equation in Minkowski space-time is (ℏ = 𝑐 = 1) [25] (𝑖𝛾𝑎 𝜕𝑎 − 𝑚) 𝜓 = 0, (1) where the 4 × 4 constant matrices 𝛾𝑎 satisfy the following anticommutation relation in terms of Minkowski metric: {𝛾𝑎 , 𝛾𝑏 } = 2𝐼𝜂𝑎 𝑏 . (2) [𝑖𝛾𝜇 (𝜕𝜇 + 1 8 𝜔𝜇 𝑐𝑑 [𝛾𝑐 , 𝛾𝑑 ]) − 𝑚] 𝜓 = 0. (8) By considering the polar coordinates (𝑟, 𝜃) the metric tensor in (2 + 1) dimension can be represented as 𝑔𝜇 ] = 𝑑𝑡 ⊗ 𝑑𝑡 − 𝑑𝑟 ⊗ 𝑑𝑟 − 𝑟2 𝑑𝜃 ⊗ 𝑑𝜃, (9) and the tetrads corresponding to this metric are as follows: 𝑒𝜇 𝑎 = (𝑒𝑎 𝜇) − 1 = ( 1 0 0 0 1 0 0 0 1 𝑟 ) . (10) The (2 + 1) Dirac equation in the presence of electromagnetic field according to metric (9) is [𝑖𝛾𝜇 (𝜕𝜇 − 𝑖𝑒𝐴 𝜇 + 1 8 𝜔𝜇 𝑐𝑑 [𝛾𝑐 , 𝛾𝑑 ]) − 𝑚] 𝜓 (𝑡, 𝑟, 𝜃) = 0, (11)

The Dirac equation in Minkowski space-time is (ℏ = 𝑐 = 1) [25] (𝑖𝛾𝑎 𝜕𝑎 − 𝑚) 𝜓 = 0, (1) where the 4 × 4 constant matrices 𝛾𝑎 satisfy the following anticommutation relation in terms of Minkowski metric: {𝛾𝑎 , 𝛾𝑏 } = 2𝐼𝜂𝑎 𝑏 . (2) Generalization of the Dirac equation from flat to c urved space-time needs two modifications. First of all, the 𝛾𝑎 matri- ces should be replaced by coordinate dependent matrices 𝛾𝜇 by using bein or tetrad formalism: 𝛾𝜇 = 𝑒𝜇 𝑎𝛾 𝑎 , (3) where the tetrads 𝑒𝑎 𝜇 and their inverse 𝑒𝜇 𝑎 transform the physical quantities from flat to curved frames and vice versa. So the metric tensor of the curved space-time can be related to Minkowski metric by 𝑔𝜇 ] = 𝜂𝑎 𝑏𝑒 𝑎 𝜇𝑒 𝑏 ]. (4) As the second modification, one should put the covariant derivative instead of partial derivative since the partial deriva- tive of a spinor does not transform like a spinor. The covariant derivative of a spinor has the following form [26]: ∇𝜇 𝜓 = 𝜕𝜇 𝜓 + Γ𝜇 𝜓 , (5) 𝑒𝜇 𝑎 = (𝑒𝑎 𝜇) − 1 = ( 1 0 0 0 1 0 0 0 1 𝑟 ) . (10) The (2 + 1) Dirac equation in the presence of electromagnetic field according to metric (9) is [𝑖𝛾𝜇 (𝜕𝜇 − 𝑖𝑒𝐴 𝜇 + 1 8 𝜔𝜇 𝑐𝑑 [𝛾𝑐 , 𝛾𝑑 ]) − 𝑚] 𝜓 (𝑡, 𝑟, 𝜃) = 0, (11) where 𝐴 𝜇 are the components of electromagnetic potential. As a convention, the Greek and Latin indices refer to global and local space-time indices which take the values 𝑡, 𝑟, 𝜃 and 0, 1, 2, respectively. In (2 + 1) dimension, the 4 × 4 Dirac matrices can be replaced by Pauli matrices: 𝛾0 = 𝜎3 , 𝛾1 = 𝑖𝜎2 , 𝛾3 = −𝑖𝜎1 , (12) which satisfy (2). By calculating the nonzero components of Christoffel symbols as Γ𝑟𝜃 𝜃 = −𝑟, Γ𝜃𝑟 𝜃 = 1 𝑟 , (13) the nonvanishing components of spin connection can be obtained according to (7) as 𝜔𝜃 12 = −𝜔𝜃 21 = 1. (14) By substituting these expressions into (11), we have

1682, 2015, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2015/612757, Wiley Online Library on [30/08/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 2

3

( 𝑖𝜕𝑡 + 𝑒𝐴 𝑡 − 𝑚 𝑖𝜕𝑟 + 1 𝑟 𝜕𝜃 − 𝑖𝑒 𝑟 𝐴 𝜃 + 𝑖 2𝑟 + 𝑒𝐴 𝑟 −𝑖𝜕𝑟 + 1 𝑟 𝜕𝜃 − 𝑖𝑒 𝑟 𝐴 𝜃 − 𝑖 2𝑟 − 𝑒𝐴 𝑟 −𝑖𝜕𝑡 − 𝑒𝐴 𝑡 − 𝑚 ) 𝜓 (𝑡, 𝑟, 𝜃) = 0.

Now, we suppose that the components of the electromagnetic potential only depend on the parameter 𝑟; hence, the tempoin terms of 𝐴 𝑡 . Therefore, by letting (18) and (20) into (17), we

potential only depend on the parameter 𝑟; hence, the tempo- ral and angular parts of the wave function can be separated as 𝜓 (𝑡, 𝑟, 𝜃) = exp [𝑖 (𝑘𝜃 − 𝐸𝑡 + 𝑒 ∫ 𝐴 𝑟𝑑 𝑟)] 𝑟− 1/2 ( 𝜑1 (𝑟) 𝑖𝜑2 (𝑟)) , (16) where the time evolution of the wave function is considered as exp(−𝑖𝐸𝑡) and 𝑘 is a constant. In (16), the terms 𝑟− 1/2 and exp(𝑒 ∫ 𝐴 𝑟𝑑 𝑟) are used in order to eliminate the factors ±(𝑖/2𝑟 + 𝑒𝐴 𝑟) from the off-diagonal elements of matrix (15). Replacing this expression in (15) yields ( −𝑒𝐴 𝑡 + 𝑚 𝑑 𝑑𝑟 + 𝑘 𝑟 − 𝑒 𝑟 𝐴 𝜃 − 𝑑 𝑑𝑟 + 𝑘 𝑟 − 𝑒 𝑟 𝐴 𝜃 −𝑒𝐴 𝑡 − 𝑚 ) (𝜑 1 (𝑟) 𝜑2 (𝑟)) = 𝐸(𝜑 1 (𝑟) 𝜑2 (𝑟)) . (17) As we see, the off-diagonal elements of the above matrix are well-known operators in supersymmetric quantum mechan- ics. This equation gives two coupled first order differential equations for two radial spinor components 𝜑1 ( 𝑟) and 𝜑2 ( 𝑟). Our calculations show that, by eliminating the lower or upper component, one can obtain a second order differential equation in terms of another which includes first order derivative. In order to get a Schrodinger-like differential equation, let us use the following unitary transformation introduced in [27, 28] (𝜑 1 (𝑟) 𝜑2 (𝑟)) =( cos 𝜌 sin 𝜌 − sin 𝜌 cos 𝜌) (𝐹 (𝑟) 𝐺 (𝑟)) , (18) where 𝜌 is a constant. After calculation, we find that the necessary condition that vanishes the first order derivative is as follows: 𝑒𝐴 𝑡 + 𝑚 cos 2𝜌 − sin 2𝜌 𝑟 (𝑘 − 𝑒𝐴 𝜃) + 𝐸 = constant. (19) get (−2𝑒𝐴 𝑡 + 𝑚 cos 2𝜌 − 𝐸) 𝐹 + 𝑑𝐺 𝑑𝑟 + (𝑚 sin 2𝜌 + 𝑒 𝑇 𝐴 𝑡) 𝐺 = 0, (𝑚 sin 2𝜌 + 𝑒 𝑇 𝐴 𝑡) 𝐹 − 𝑑𝐹 𝑑𝑟 − (𝑚 cos 2𝜌 + 𝐸) 𝐺 = 0, (21) where 𝑇 = tan 2𝜌. By using these equations, we obtain the following Schrodinger-like second order differential equation for 𝐹(𝑟): − 𝑑2 𝐹 𝑑𝑟2 +( 𝑒 𝑇 𝑑𝐴 𝑡 𝑑𝑟 − 2𝑒𝐸𝐴 𝑡 + 𝑒2 𝐴2 𝑡 𝑇2 − 𝐸2 + 𝑚2 ) 𝐹 = 0. (22) Also the lower component 𝐺(𝑟) can be written in terms of upper component as 𝐺 (𝑟) = 1 𝐸 + 𝑚 cos 2𝜌 (− 𝑑 𝑑𝑟 + 𝑚 sin 2𝜌 + 𝑒𝐴 𝑡 𝑇 ) 𝐹 (𝑟) . (23) In [29], we have solved (22) for two different kinds of electrostatic potentials. Here we want to extend our work and give general formalism in order to solve this equation for different kinds of electrostatic potentials. It is easy to see that the second term of (22) can be written as Riccati equation, and so using the properties of supersymmetry in quantum mechanics, the electrostatic potential 𝐴 𝑡 can be related to super partner of solvable potentials. Hence, we want to find the relevant choices for 𝐴 𝑡, such that (22) can be factorized into the product of two first order differential operators. Of course, we use the master function formalism introduced in [19, 20] and try to relate the shape invariance and supersymmetric properties of second order differential equations of mathematical physics to our work. So, in the next section, we first review this formalism briefly.

𝐴 𝜃 = 𝑘 𝑒 − 𝑟 sin 2𝜌 𝐴 𝑡. (20)

3. Shape Invariance Property of Second Order Differential Equations of Mathematical Physics: Master Function Approach

The general form of second order differential equation can be written in terms of master function 𝐴(𝑥) which is 1682, 2015, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2015/612757, Wiley Online Library on [30/08/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Advances in High Energy Physics

Advances in High Energy Physics a polynomial at most order two and corresponding weight

a polynomial at most order two and c function 𝑊(𝑥) as [20] 𝐴 (𝑥) 𝜑󸀠 󸀠 𝑛 (𝑥) + (𝐴 (𝑥) 𝑊 (𝑥))󸀠 𝑊 (𝑥) 𝜑󸀠𝑛 (𝑥)

̃ ̃ From (27), it is evident that 𝐵(𝑛) and 𝐴(𝑛), as the raising and lowering operators, respectively, satisfy the following relations:

𝐴 (𝑥) 𝜑󸀠 󸀠 𝑛 (𝑥) + (𝐴 (𝑥) 𝑊 (𝑥))󸀠 𝑊 (𝑥) 𝜑󸀠𝑛 (𝑥) − [𝑛 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 + 𝑛 (𝑛 + 1) 2 𝐴󸀠 󸀠 (𝑥)] 𝜑𝑛 (𝑥) = 0, (24) relations: ̃𝐵 (𝑛) 𝜑𝑛 −1 (𝑥) = √𝐸 (𝑛)𝜑𝑛 (𝑥) , ̃𝐴 (𝑛) 𝜑𝑛 (𝑥) = √𝐸 (𝑛)𝜑𝑛 −1 (𝑥) . (29) As we see, the operators ̃𝐵(𝑛) and ̃𝐴(𝑛) in (26) are not the differential supersymmetric self-adjoint operators. By

𝐴 (𝑥) 𝜑𝑛 (𝑥) + 𝑊 (𝑥) 𝜑𝑛 (𝑥) − [𝑛 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 + 𝑛 (𝑛 + 1) 2 𝐴󸀠 󸀠 (𝑥)] 𝜑𝑛 (𝑥) = 0, (24) where the nonnegative weight function 𝑊(𝑥) at the inter- val [𝑎, 𝑏] is calculated so that (1/𝑊(𝑥))(𝑑/𝑑𝑥)(𝐴(𝑥)𝑊(𝑥)) becomes at most first order polynomial. The interval [𝑎, 𝑏] is appointed in this manner that 𝐴(𝑎)𝑊(𝑎) = 𝐴(𝑏)𝑊(𝑏) = 0. The polynomials 𝜑𝑛 ( 𝑥) have the following Rodrigues representation: 𝜑𝑛 (𝑥) = 𝑎𝑛 𝑊 (𝑥) ( 𝑑 𝑑𝑥 ) 𝑛 (𝐴𝑛 (𝑥) 𝑊 (𝑥)) . (25) By introducing two first order differential operators ̃𝐵 (𝑛) = 𝐴 (𝑥) 𝑑 𝑑𝑥 + 𝑛 2 𝐴󸀠 (𝑥) + 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) +𝑛 ⋅ 𝐴󸀠 (0) (𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 − 𝐴󸀠 󸀠 (𝑥) (𝐴𝑊󸀠 /𝑊) (0) 2 [(𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥)] , ̃𝐴 (𝑛) = −𝐴 (𝑥) 𝑑 𝑑𝑥 + 𝑛 2 𝐴󸀠 (𝑥) + 𝑛 ⋅ 𝐴󸀠 (0) (𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 − 𝐴󸀠 󸀠 (𝑥) (𝐴𝑊󸀠 /𝑊) (0) 2 [(𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥)] , (26) (24) can be factorized in a shape invariant form with respect to quantum number 𝑛: ̃𝐵 (𝑛) ̃𝐴 (𝑛) 𝜑𝑛 (𝑥) = 𝐸 (𝑛) 𝜑𝑛 (𝑥) , ̃𝐴 (𝑛) ̃𝐵 (𝑛) 𝜑𝑛 −1 (𝑥) = 𝐸 (𝑛) 𝜑𝑛 −1 (𝑥) , (27) where the energy spectrum is given by 𝐸 (𝑛) = 𝑛 4 [(𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥)]2 {4 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠2 ⋅ [𝑛𝐴󸀠 2 (0) − 𝐴 (0) ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 ]−( 𝐴𝑊󸀠 𝑊 ) (0) ⋅ [𝐴󸀠 󸀠 (𝑥) ( 𝐴𝑊󸀠 𝑊 ) (0) − 2𝐴󸀠 (0) ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 ] ⋅ [2 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥)] + 𝑛2 𝐴󸀠 󸀠 (0) ⋅ (𝐴󸀠 2 (0) − 2𝐴󸀠 󸀠 (0) 𝐴 (0)) [𝑛𝐴󸀠 󸀠 (0) + 4 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 ] − 10𝑛𝐴 (0) 𝐴󸀠 󸀠 (𝑥) ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) ) 󸀠2 }. (28) ̃𝐵 (𝑛) 𝜑𝑛 −1 (𝑥) = √𝐸 (𝑛)𝜑𝑛 (𝑥) , ̃𝐴 (𝑛) 𝜑𝑛 (𝑥) = √𝐸 (𝑛)𝜑𝑛 −1 (𝑥) . (29) As we see, the operators ̃𝐵(𝑛) and ̃𝐴(𝑛) in (26) are not the differential supersymmetric self-adjoint operators. By inserting a similarity transformation and a change of variable as 𝐴 (𝑛) = 𝑊1 /2 (𝑥) ̃𝐵 (𝑛) 𝑊− 1/2 (𝑥) , 𝐴† (𝑛) = 𝑊1 /2 (𝑥) ̃𝐴 (𝑛) 𝑊− 1/2 (𝑥) , (30) 𝑑𝑟 𝑑𝑥 = 1 𝐴 (𝑥) , (31) (27) can be rewritten as the following form: 𝐴 (𝑛) 𝐴† (𝑛) 𝜓𝑛 (𝑟) = 𝐸 (𝑛) 𝜓𝑛 (𝑟) , (32) 𝐴† (𝑛) 𝐴 (𝑛) 𝜓𝑛 −1 (𝑟) = 𝐸 (𝑛) 𝜓𝑛 −1 (𝑟) . (33) The new self-adjoint operators 𝐴(𝑛) and 𝐴† (𝑛) are given by 𝐴 (𝑛) = 𝑑 𝑑𝑟 + 𝑤𝑛 (𝑟) , 𝐴† (𝑛) = − 𝑑 𝑑𝑟 + 𝑤𝑛 (𝑟) , (34) where the superpotential 𝑤𝑛 ( 𝑟) can be determined as 𝑤𝑛 (𝑟) = 1 2 [ [ 𝑛 𝐴󸀠 (𝑥) + 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) +𝑛 ⋅ 𝐴󸀠 (0) (𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 − 𝐴󸀠 󸀠 (𝑥) (𝐴𝑊󸀠 /𝑊) (0) (𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥) ] ]𝑥 =𝑥(𝑟) (35) and the wave function 𝜓𝑛 ( 𝑟) can be obtained by product of 𝑊1 /2 (𝑥) into orthogonal polynomials 𝜑𝑛 ( 𝑥) as 𝜓𝑛 (𝑟) = 𝑊1 /2 (𝑥) 𝜑𝑛 (𝑥) = 𝑎𝑛 𝑊1 /2 (𝑥) ( 𝑑 𝑑𝑥 ) 𝑛 (𝐴𝑛 (𝑥) 𝑊 (𝑥)) 󵄨 󵄨󵄨󵄨󵄨󵄨 󵄨 󵄨𝑥 =𝑥(𝑟) . (36) Also from (29), (30), and (36), one can get 𝐴 (𝑛) 𝜓𝑛 −1 (𝑟) = √𝐸 (𝑛)𝜓𝑛 (𝑟) , † (37)

𝑛 −1 𝑛 −1 where the energy spectrum is given by 𝐸 (𝑛) = 𝑛 4 [(𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥)]2 {4 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠2 ⋅ [𝑛𝐴󸀠 2 (0) − 𝐴 (0) ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 ]−( 𝐴𝑊󸀠 𝑊 ) (0) ⋅ [𝐴󸀠 󸀠 (𝑥) ( 𝐴𝑊󸀠 𝑊 ) (0) − 2𝐴󸀠 (0) ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 ] ⋅ [2 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥)] + 𝑛2 𝐴󸀠 󸀠 (0) ⋅ (𝐴󸀠 2 (0) − 2𝐴󸀠 󸀠 (0) 𝐴 (0)) [𝑛𝐴󸀠 󸀠 (0) + 4 ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠 ] − 10𝑛𝐴 (0) 𝐴󸀠 󸀠 (𝑥) ( 𝐴 (𝑥) 𝑊󸀠 (𝑥) 𝑊 (𝑥) ) 󸀠2 }. (28) [ ⋅ 𝐴󸀠 (0) (𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 − 𝐴󸀠 󸀠 (𝑥) (𝐴𝑊󸀠 /𝑊) (0) (𝐴 (𝑥) 𝑊󸀠 (𝑥) /𝑊 (𝑥))󸀠 + 𝑛𝐴󸀠 󸀠 (𝑥) ] ]𝑥 =𝑥(𝑟) (35) and the wave function 𝜓𝑛 ( 𝑟) can be obtained by product of 𝑊1 /2 (𝑥) into orthogonal polynomials 𝜑𝑛 ( 𝑥) as 𝜓𝑛 (𝑟) = 𝑊1 /2 (𝑥) 𝜑𝑛 (𝑥) = 𝑎𝑛 𝑊1 /2 (𝑥) ( 𝑑 𝑑𝑥 ) 𝑛 (𝐴𝑛 (𝑥) 𝑊 (𝑥)) 󵄨 󵄨󵄨󵄨󵄨󵄨 󵄨 󵄨𝑥 =𝑥(𝑟) . (36) Also from (29), (30), and (36), one can get 𝐴 (𝑛) 𝜓𝑛 −1 (𝑟) = √𝐸 (𝑛)𝜓𝑛 (𝑟) , 𝐴† (𝑛) 𝜓𝑛 (𝑟) = √𝐸 (𝑛)𝜓𝑛 −1 (𝑟) . (37)

1682, 2015, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2015/612757, Wiley Online Library on [30/08/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4

5

4. Solution of the Curved Dirac Equation via Master Function Approach

In this section, we want to determine the electrostatic potential such that (22) becomes Schrodinger equation with some shape invariant potentials. As a result, the relativistic energy and wave functions of the curved Dirac equation can be obtained in terms of nonrelativistic energy and wave functions. For this aim, we insert the operators 𝐴(𝑛) and 𝐴† (𝑛) given in (34) into Schrodinger equation (32) and get [− 𝑑2 𝑑𝑟2 + 𝑤2𝑛 (𝑟) + 𝑑𝑤𝑛 (𝑟) 𝑑𝑟 ] 𝜓𝑛 (𝑟) = 𝐸 (𝑛) 𝜓𝑛 (𝑟) . (38) Let us compare (22) with the above Schrodinger equation derived from master function formalism 𝑤2𝑛 (𝑟) + 𝑑𝑤𝑛 (𝑟) 𝑑𝑟 − 𝐸 (𝑛) = 𝑒 𝑇 𝑑𝐴 𝑡 𝑑𝑟 − 2𝑒𝐸𝐴 𝑡 + 𝑒2 𝐴2 𝑡 𝑇2 − 𝐸2 + 𝑚2 . (39) Solving this equation gives the electrostatic potential and relativistic energy as 𝐴 𝑡 = 𝑇 𝑒 (𝑤𝑛 (𝑟) + 𝐸𝑇) , (40) 𝐸 = cos 2𝜌 [𝐸 (𝑛) + 𝑚2 ]1 /2 . (41) Therefore, by letting different choices for master function 𝐴(𝑥) and weight function 𝑊(𝑥), we specify the superpoten- tial and nonrelativistic energy according to (35) and (28); then, by substituting them into (40) and (41), we obtain the electrostatic potential and relativistic energy which satisfy the Dirac equation (17). The upper spinor component is deter- mined in terms of Rodrigues representation of orthogonal polynomials from (36). Also, we can calculate the lower component of the wave function from (23) and (37) as 𝐺 (𝑟) = 1 𝐸 + 𝑚 cos 2𝜌 (√𝐸 (𝑛)𝜓𝑛 −1 (𝑟) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝜓𝑛 (𝑟)) . (42) Finally, according to condition (20), the angular part of electromagnetic potential is obtained as 𝐴 𝜃 = 𝑘 𝑒 − 𝑟 sin 2𝜌 𝑇 𝑒 (𝑤𝑛 (𝑟) + 𝐸𝑇) . (43) 𝐹 = 𝑑𝐴 = 𝑇 𝑒 𝑑𝑤𝑛 (𝑟) 𝑑𝑟 𝑑𝑟 ∧ 𝑑𝑡 − 1 sin 2𝜌 𝑇 𝑒 [𝑤𝑛 (𝑟) + 𝐸𝑇 + 𝑟 𝑑𝑤𝑛 (𝑟) 𝑑𝑟 ] 𝑑𝑟 ∧ 𝑑𝜃. (45) It is obvious that the radial part of electromagnetic potential 𝐴 𝑟 does not contribute to the magnetic field because of its radial symmetry. In the following, we apply this formalism for a special case of master function and the other cases are given in the Appendix. Example 1. Let 𝐴(𝑥) = 𝑥. Then, taking into consideration that (1/𝑊(𝑥))(𝑑/𝑑𝑥)(𝐴(𝑥)𝑊(𝑥)) is at most first order poly- nomial, the weight function 𝑊(𝑥) can be obtained as 𝑊 (𝑥) = 𝑥𝛼 𝑒− 𝛽𝑥 , 0 ≤ 𝑥 < +∞, 𝛼 > −1, 𝛽 > 0. (46) Using (35), the superpotential is determined as follows: 𝑤𝑛 (𝑟) = 1 2 (2𝑛 + 𝛼 − 𝛽𝑥) , (47) where by changing the variable 𝑥 = 𝑒𝑟 , according to (31), it leads to superpotential of the Morse potential [31, 32]. Now we determine the electrostatic potential of the Dirac equation from (40) 𝐴 𝑡 = 𝑇 𝑒 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] . (48) Substituting this expression into (22) yields − 𝑑2 𝐹 (𝑟) 𝑑𝑟2 + ( 𝛽2 4 𝑒2 𝑟 − 𝛽 2 𝑒𝑟 (1 + 2𝑛 + 𝛼) + 1 4 (2𝑛 + 𝛼)2 − 𝐸2 (𝑇2 + 1) + 𝑚2 ) 𝐹 (𝑟) = 0, (49) where by letting the relativistic energy according to (41) and (28) 𝐸 = cos 2𝜌 [𝑛 (𝑛 + 𝛼) + 𝑚2 ]1 /2 ; (50) then it can be factorized as the product of raising and lowering operators

= 𝑒 𝑇 𝑑𝐴 𝑡 𝑑𝑟 − 2𝑒𝐸𝐴 𝑡 + 𝑒 𝐴 𝑡 𝑇2 − 𝐸2 + 𝑚2 . Solving this equation gives the electrostatic potential and relativistic energy as 𝐴 𝑡 = 𝑇 𝑒 (𝑤𝑛 (𝑟) + 𝐸𝑇) , (40) 𝐸 = cos 2𝜌 [𝐸 (𝑛) + 𝑚2 ]1 /2 . (41) Therefore, by letting different choices for master function 𝐴(𝑥) and weight function 𝑊(𝑥), we specify the superpoten- tial and nonrelativistic energy according to (35) and (28); then, by substituting them into (40) and (41), we obtain the electrostatic potential and relativistic energy which satisfy the Dirac equation (17). The upper spinor component is deter- mined in terms of Rodrigues representation of orthogonal polynomials from (36). Also, we can calculate the lower component of the wave function from (23) and (37) as 𝐺 (𝑟) = 1 𝐸 + 𝑚 cos 2𝜌 (√𝐸 (𝑛)𝜓𝑛 −1 (𝑟) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝜓𝑛 (𝑟)) . (42) Finally, according to condition (20), the angular part of electromagnetic potential is obtained as 𝐴 𝜃 = 𝑘 𝑒 − 𝑟 sin 2𝜌 𝑇 𝑒 (𝑤𝑛 (𝑟) + 𝐸𝑇) . (43) On the other hand, we can specify the electromagnetic tensor. It is well known that the electromagnetic potential is a one form [30] 𝐴 = 𝐴 𝜇𝑑 𝑥𝜇 = 𝐴 𝑡𝑑 𝑡 + 𝐴 𝑟𝑑 𝑟 + 𝐴 𝜃𝑑 𝜃 = 𝑇 𝑒 (𝑤𝑛 (𝑟) + 𝐸𝑇) 𝑑𝑡 + 𝐴 𝑟 (𝑟) 𝑑𝑟 +( 𝑘 𝑒 − 𝑟 sin 2𝜌 𝑇 𝑒 (𝑤𝑛 (𝑟) + 𝐸𝑇)) 𝑑𝜃. (44) nomial, the weight function 𝑊(𝑥) can be obtained as 𝑊 (𝑥) = 𝑥𝛼 𝑒− 𝛽𝑥 , 0 ≤ 𝑥 < +∞, 𝛼 > −1, 𝛽 > 0. (46) Using (35), the superpotential is determined as follows: 𝑤𝑛 (𝑟) = 1 2 (2𝑛 + 𝛼 − 𝛽𝑥) , (47) where by changing the variable 𝑥 = 𝑒𝑟 , according to (31), it leads to superpotential of the Morse potential [31, 32]. Now we determine the electrostatic potential of the Dirac equation from (40) 𝐴 𝑡 = 𝑇 𝑒 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] . (48) Substituting this expression into (22) yields − 𝑑2 𝐹 (𝑟) 𝑑𝑟2 + ( 𝛽2 4 𝑒2 𝑟 − 𝛽 2 𝑒𝑟 (1 + 2𝑛 + 𝛼) + 1 4 (2𝑛 + 𝛼)2 − 𝐸2 (𝑇2 + 1) + 𝑚2 ) 𝐹 (𝑟) = 0, (49) where by letting the relativistic energy according to (41) and (28) 𝐸 = cos 2𝜌 [𝑛 (𝑛 + 𝛼) + 𝑚2 ]1 /2 ; (50) then it can be factorized as the product of raising and lowering operators 𝐴 (𝑛) 𝐴† (𝑛) 𝐹𝑛 (𝑟) = 𝑛 (𝑛 + 𝛼) 𝐹𝑛 (𝑟) , 𝐴† (𝑛) 𝐴 (𝑛) 𝐹𝑛 −1 (𝑟) = 𝑛 (𝑛 + 𝛼) 𝐹𝑛 −1 (𝑟) , (51) where 𝐴 (𝑛) = 𝑑 𝑑𝑟 + 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) , 𝐴† (𝑛) = − 𝑑 𝑑𝑟 + 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) . (52)

1682, 2015, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2015/612757, Wiley Online Library on [30/08/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Advances in High Energy Physics

Advances in High Energy Physics Introducing a new function according to (36) as

𝐹𝑛 (𝑥 (𝑟)) = 𝑥𝛼 /2 𝑒− (𝛽/2)𝑥 𝜑𝑛 (𝑥 (𝑟)) ,

for the curved Dirac equation in polar coordinates have been obtained and in the following we give them briefly: (53)

and changing variable 𝑑𝑟/𝑑𝑥 = 1/𝑥, (49) is transformed into the following differential equation: 𝑥𝜑󸀠 󸀠 𝑛 (𝑥) + [(𝛼 + 1) − 𝛽𝑥] 𝜑󸀠𝑛 (𝑥) + 𝑛𝛽𝜑𝑛 (𝑥) = 0, (54) 𝐴 (𝑥) = 1, 𝑊 (𝑥) = 𝑒− (1/2)𝛼𝑥2 +𝛽𝑥 − ∞ < 𝑥 < +∞, 𝛼 > 0, −∞ < 𝛽 < +∞, 𝑟 = 𝑥,

𝑥𝜑𝑛 (𝑥) + [(𝛼 + 1) − 𝛽𝑥] 𝜑𝑛 (𝑥) + 𝑛𝛽𝜑𝑛 (𝑥) = 0, (54) which is the associated Laguerre differential equation where 𝜑𝑛 ( 𝑥) is specified from (25) as 𝜑𝑛 (𝑥) = 𝑎𝑛 𝑥 − 𝛼 𝑒𝛽 𝑥 ( 𝑑 𝑑𝑥 ) 𝑛 (𝑥𝑛 +𝛼 𝑒− 𝛽𝑥 ) . (55) 𝐴 𝑡 = 𝑇 𝑒 (− 𝛼 2 𝑟 + 𝛽 2 + 𝐸𝑇) , 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟 (− 𝛼 2 𝑟 + 𝛽 2 + 𝐸𝑇) , 𝐹=− 𝛼𝑇 2𝑒 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 (−𝛼𝑟 + 𝛽 2 + 𝐸𝑇) 𝑑𝑟 ∧ 𝑑𝜃,

Also from (42), we get 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 [√𝑛 (𝑛 + 𝛼)𝐹𝑛 −1 (𝑥 (𝑟)) (56) 2𝑒 𝑒 sin 2𝜌 2 𝐸 = cos 2𝜌 [𝑛𝛼 + 𝑚2 ]1 /2 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 𝑒 ( 1/4)𝛼𝑥2 −(𝛽/2)𝑥 ( 𝑑 𝑑𝑥 ) 𝑛 (𝑒− (1/2)𝛼𝑥2 +𝛽𝑥 ),

𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 [√𝑛 (𝑛 + 𝛼)𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))] . (56) 𝐸 = cos 2𝜌 [𝑛𝛼 + 𝑚 ] 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 𝑒 ( 1/4)𝛼𝑥2 −(𝛽/2)𝑥 ( 𝑑 𝑑𝑥 ) 𝑛 (𝑒− (1/2)𝛼𝑥2 +𝛽𝑥 ), 𝑎𝑛 = 𝑎𝑛 −1 √𝑛𝛼 ,

𝐺 (𝑥 (𝑟)) = 𝐸 + 𝑚 cos 2𝜌 [𝑛 (𝑛 + 𝛼)𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))] . (56) According to condition (43), we have the following relation for angular component of electromagnetic potential: 𝐴 𝜃 = 𝑘 𝑒 − 𝑟 sin 2𝜌 𝑇 𝑒 ( 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇) . (57) By knowing the electromagnetic potential as a one form satisfies (44), we can calculate the electromagnetic tensor as 𝐹 = − 𝑇𝛽 2𝑒 𝑒𝑟 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇 − 𝛽 2 𝑟𝑒𝑟 ] 𝑑𝑟 (58) 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 𝑒 ( 𝑑𝑥 ) (𝑒 ), 𝑎𝑛 = 𝑎𝑛 −1 √𝑛𝛼 , 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 (√𝑛𝛼𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))) . 𝐴 (𝑥) = 𝑥, 𝑊 (𝑥) = 𝑥𝛼 𝑒− 𝛽𝑥 , 0 ≤ 𝑥 < +∞, 𝛼 > −1, 𝛽 > 0, 𝑥=𝑒𝑟 , 𝐴 𝑡 = 𝑇 𝑒 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] , 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] , 𝐸 = cos 2𝜌 [𝑛 (𝑛 + 𝛼) + 𝑚2 ]1 /2 ,

(A.1)

𝐸 = cos 2𝜌 [𝑛𝛼 + 𝑚2 ]1 /2 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 𝑒 ( 1/4)𝛼𝑥2 −(𝛽/2)𝑥 ( 𝑑 𝑑𝑥 ) 𝑛 (𝑒− (1/2)𝛼𝑥2 +𝛽𝑥 ), 𝑎𝑛 = 𝑎𝑛 −1 √𝑛𝛼 , 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 (√𝑛𝛼𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))) . 𝐴 (𝑥) = 𝑥, 𝑊 (𝑥) = 𝑥𝛼 𝑒− 𝛽𝑥 , 0 ≤ 𝑥 < +∞, 𝛼 > −1, 𝛽 > 0, 𝑥=𝑒𝑟 , 𝐴 𝑡 = 𝑇 𝑒 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] , 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] ,

𝑒 sin 2𝜌 𝑒 2 By knowing the electromagnetic potential as a one form satisfies (44), we can calculate the electromagnetic tensor as 𝐹 = − 𝑇𝛽 2𝑒 𝑒𝑟 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇 − 𝛽 2 𝑟𝑒𝑟 ] 𝑑𝑟 ∧ 𝑑𝜃. (58) 5. Conclusion In this paper, we have studied the (2 + 1) Dirac equation in curved space-time in the presence of both electric and magnetic fields. We have tried to transform this equation into a Schrodinger-like equation for the upper spinor component by applying a unitary transformation and constraint between the components of the gauge field. We have also proposed the electrostatic potentials and relativistic energies in such a way that this equation was transformed into second order differ- ential equations of mathematical physics in master function formalism. The spinor wave function is also specified in terms of Rodrigues polynomials related to each differential equation. Appendix 𝑊 (𝑥) = 𝑥𝛼 𝑒− 𝛽𝑥 , 0 ≤ 𝑥 < +∞, 𝛼 > −1, 𝛽 > 0, 𝑥=𝑒𝑟 , 𝐴 𝑡 = 𝑇 𝑒 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] , 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇] , 𝐸 = cos 2𝜌 [𝑛 (𝑛 + 𝛼) + 𝑚2 ]1 /2 , 𝐹=− 𝑇𝛽 2𝑒 𝑒𝑟 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 [ 1 2 (2𝑛 + 𝛼 − 𝛽𝑒𝑟 ) + 𝐸𝑇 − 𝛽 2 𝑟𝑒𝑟 ] 𝑑𝑟 ∧ 𝑑𝜃, 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 𝑥 − 𝛼/2 𝑒( 𝛽/2)𝑥 ( 𝑑 𝑑𝑥 ) 𝑛 (𝑥𝑛 +𝛼 𝑒− 𝛽𝑥 ) , 𝑎𝑛 = 𝑎𝑛 −1 √𝑛 (𝑛 + 𝛼) , 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 [√𝑛 (𝑛 + 𝛼)𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))] , (A.2) 𝐴 (𝑥) = 1 − 𝑥2 , 𝑊 (𝑥) = (1 − 𝑥)𝛼 (1 + 𝑥)𝛽 , −1 < 𝑥 < +1, 𝛼 > −1, 𝛽 > −1, 𝑥 = tanh 𝑟, 𝐴 𝑡 = 𝑇 𝑒 [ −2𝑛 − 𝛼 − 𝛽 2 tanh 𝑟 + 𝛽2 − 𝛼2 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] ,

Here, we list some choices for master function and corresponding weight function. For each case, similar to given example in this paper, the components of gauge field, electromagnetic tensor, energy spectra, and spinor wave functions

𝑒 2 2 (2𝑛 + 𝛼 + 𝛽) 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟[ −2𝑛 − 𝛼 − 𝛽 2 tanh 𝑟 + 𝛽2 − 𝛼2 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] ,

1682, 2015, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2015/612757, Wiley Online Library on [30/08/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6

𝐸 = cos 2𝜌 [ 4𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 + 𝑚2 ] 1/2 𝐹= −𝑇 (2𝑛 + 𝛼 + 𝛽) 2𝑒 1 cosh2 𝑟 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 [ −2𝑛 − 𝛼 − 𝛽 2 tanh 𝑟 + 𝛽2 − 𝛼2 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 (√ 4𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))) . (A.4)

𝐸 = cos 2𝜌 [ 4𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 + 𝑚2 ] 𝐹= −𝑇 (2𝑛 + 𝛼 + 𝛽) 2𝑒 1 cosh2 𝑟 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 [ −2𝑛 − 𝛼 − 𝛽 2 tanh 𝑟 + 𝛽2 − 𝛼2 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇 + −2𝑛 − 𝛼 − 𝛽 2 𝑟 cosh2 𝑟 ] 𝑑𝑟 ∧ 𝑑𝜃, 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 (1 − 𝑥)− 𝛼/2 (1 + 𝑥)− 𝛽/2 ( 𝑑 𝑑𝑥 ) 𝑛 ⋅ ((1 − 𝑥)𝑛 +𝛼 (1 + 𝑥)𝑛 +𝛽 ), 𝑎𝑛 = 1 2 √ 𝑛+𝛼+𝛽 𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) 𝑎𝑛 −1 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 [ [ √ 4𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))] ] . (A.3) = 1 𝐸 + 𝑚 cos 2𝜌 (√ 4𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))) . (A.4) 𝐴 (𝑥) = 𝑥 (1 − 𝑥) , 𝑊 (𝑥) = 𝑥𝛼 (1 − 𝑥)𝛽 , 0 < 𝑥 < +1, 𝛼 > −1, 𝛽 > −1, 𝑥= 𝑒𝑟 1 + 𝑒𝑟 , 𝐴 𝑡 = 𝑇 𝑒 [− 1 2 (2𝑛 + 𝛼 + 𝛽) 𝑒𝑟 1 + 𝑒𝑟 + (𝑛 + 𝛼) (2𝑛 + 𝛼 + 𝛽) + (𝛽 − 𝛼) 𝑛 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟 [− 1 2 (2𝑛 + 𝛼 + 𝛽) 𝑒𝑟 1 + 𝑒𝑟 + (𝑛 + 𝛼) (2𝑛 + 𝛼 + 𝛽) + (𝛽 − 𝛼) 𝑛 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] 𝐸 = cos 2𝜌 [ 𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 + 𝑚2 ] 1/2

+ −2𝑛 − 𝛼 − 𝛽 2 𝑟 cosh2 𝑟 ] 𝑑𝑟 ∧ 𝑑𝜃, 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 (1 − 𝑥)− 𝛼/2 (1 + 𝑥)− 𝛽/2 ( 𝑑 𝑑𝑥 ) 𝑛 ⋅ ((1 − 𝑥)𝑛 +𝛼 (1 + 𝑥)𝑛 +𝛽 ), 𝑎𝑛 = 1 2 √ 𝑛+𝛼+𝛽 𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) 𝑎𝑛 −1 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 [ [ √ 4𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))] ] . (A.3) A (𝑥) = 𝑥2 − 1, 𝑊 (𝑥) = (𝑥 − 1)𝛼 (𝑥 + 1)𝛽 , − ∞ < 𝑥 < −1, 𝛽 > −1, 𝛼 + 𝛽 < −2, 𝑥 = − coth 𝑟, 𝐴 𝑡 = 𝑇 𝑒 [− 2𝑛 + 𝛼 + 𝛽 2 coth 𝑟 + 𝛼2 − 𝛽2 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] , 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟 [− 2𝑛 + 𝛼 + 𝛽 2 coth 𝑟 + 𝛼2 − 𝛽2 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] , 𝐸 = cos 2𝜌 [ 4𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 + 𝑚2 ] 1/2 𝐹= 𝑇 (2𝑛 + 𝛼 + 𝛽) 2𝑒 1 sinh2 𝑟 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 [− 2𝑛 + 𝛼 + 𝛽 2 coth 𝑟 + 𝛼2 − 𝛽2 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇 𝐴 (𝑥) = 𝑥 (1 − 𝑥) , 𝑊 (𝑥) = 𝑥𝛼 (1 − 𝑥)𝛽 , 0 < 𝑥 < +1, 𝛼 > −1, 𝛽 > −1, 𝑥= 𝑒𝑟 1 + 𝑒𝑟 , 𝐴 𝑡 = 𝑇 𝑒 [− 1 2 (2𝑛 + 𝛼 + 𝛽) 𝑒𝑟 1 + 𝑒𝑟 + (𝑛 + 𝛼) (2𝑛 + 𝛼 + 𝛽) + (𝛽 − 𝛼) 𝑛 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] 𝐴 𝜃 = 𝑘 𝑒 − 𝑇 𝑒 sin 2𝜌 𝑟 [− 1 2 (2𝑛 + 𝛼 + 𝛽) 𝑒𝑟 1 + 𝑒𝑟 + (𝑛 + 𝛼) (2𝑛 + 𝛼 + 𝛽) + (𝛽 − 𝛼) 𝑛 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇] 𝐸 = cos 2𝜌 [ 𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 + 𝑚2 ] 1/2 𝐹=− 𝑇 2𝑒 (2𝑛 + 𝛼 + 𝛽) 𝑒𝑟 (1 + 𝑒𝑟 ) 2 𝑑𝑟 ∧ 𝑑𝑡 − 𝑇 𝑒 sin 2𝜌 [− 1 2 (2𝑛 + 𝛼 + 𝛽) 𝑒𝑟 1 + 𝑒𝑟 + (𝑛 + 𝛼) (2𝑛 + 𝛼 + 𝛽) + (𝛽 − 𝛼) 𝑛 2 (2𝑛 + 𝛼 + 𝛽) + 𝐸𝑇 − 1 2 (2𝑛 + 𝛼 + 𝛽) 𝑟𝑒𝑟 (1 + 𝑒𝑟 ) 2 ] 𝑑𝑟 ∧ 𝑑𝜃, 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 𝑥 − 𝛼/2 (1 − 𝑥)− 𝛽/2 ( 𝑑 𝑑𝑥 ) 𝑛 (𝑥𝑛 +𝛼 (1 − 𝑥)𝑛 +𝛽 ) , 𝑎𝑛 = √ 𝑛+𝛼+𝛽 𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) 𝑎𝑛 −1, 𝐺 (𝑥 (𝑟)) = 1 𝐸 + 𝑚 cos 2𝜌 (√ 𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽)2 𝐹𝑛 −1 (𝑥 (𝑟)) + (𝑚 sin 2𝜌 + 𝐸𝑇) 𝐹𝑛 (𝑥 (𝑟))) . (A.5) Conflict of Interests

sinh 𝑟 𝐹𝑛 (𝑥 (𝑟)) = 𝑎𝑛 (𝑥 − 1)− 𝛼/2 (𝑥 + 1)− 𝛽/2 ( 𝑑 𝑑𝑥 ) 𝑛 ⋅ ((𝑥 − 1)𝑛 +𝛼 (𝑥 + 1)𝑛 +𝛽 ) , 𝑎𝑛 = 1 2 √ 𝑛+𝛼+𝛽 𝑛 (𝑛 + 𝛼) (𝑛 + 𝛽) 𝑎𝑛 −1

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