Comment on “Dirac fermions in Som–Raychaudhuri space-time with scalar and vector potential and the energy momentum distributions [Eur. Phys. J. C (2019) 79:541]”

Francisco A. Cruz Neto1, Andrés G. Jirón Vicente1,2, Luis B. Castro1,3,a 1 Departamento de Física, Universidade Federal do Maranhão (UFMA), Campus Universitário do Bacanga, 65080-805 São Luís, MA, Brazil 2 Universidad Tecnológica del Perú (UTP), Av. N. Ayllon Km 11.60, Lt 96, ATE, Lima, Peru 3 Departamento de Física e Química, Universidade Estadual Paulista (UNESP), Campus de Guaratinguetá, 12516-410 Guaratinguetá, SP, Brazil

In a recent paper in this Journal, Sedaghatnia et al. [1] have studied the Dirac equation in the presence of scalar and vector potentials in a class of flat Gödel-type space-times called Som-Raychaudhuri space-times by using the methods quasiexactly solvable (QES) differential equations and the Nikiforov Uvarov (NU) form. To achieve their goal, the authors have mapped the system into second-order differential equation (Schrödinger-like problem). It is worthwhile to mention that the expressions obtained [Eqs. (2.8)-(2.17)] in Ref. [1] are correct. On the other hand, the second-order differential equation in Ref. [1] is wrong, probably due to erroneous calculations in the manipulation of the two coupled first-order differential equations. This fact jeopardizes the results of [1]. The purpose of this comment is to calculate the correct differential equation and following the appropriate procedure to obtain the solution for this problem.
The Gödel-type solution with torsion and a topological defect can be written in cylindrical coordinates by the line element The Dirac equation for a free Fermi field Ψ of mass M in a Som-Raychaudhuri space-time with scalar and vector potentials is given by [1] where A μ = (V (r ), 0, 0, 0), ∇ μ = (∂ μ + Γ μ ) and Γ μ is the affine connection. Now, using the correct results of Ref. [1] and considering the solution in the form the Dirac equation (2) Using the expression forχ(r ) obtained from (5) with V (r ) = S(r ), redefining the spinor asψ(r ) = ψ(r ) √ r and inserting it in (4) we obtain Equation (6) is effectively a Schrödinger-type equation. The second-order differential equation obtained in [1] [Eq. (2.18)] is not similar to our result (6) probably due to erroneous calculations in the manipulation of the two coupled firstorder differential Eqs. (4) and (5). As in Ref. [1], firstly we concentrate our efforts on V (r ) = and V (r ) = 0, (6) reduces to where The equation of motion (7) describes the quantum dynamics of a Dirac particle in a Som-Raychaudhuri space-time. The expression for λ 2 obtained in Ref. [1] [Eq. (2.21)] is wrong. The solution for (7) with λ 1 and λ 3 real is precisely the wellknown solution of the Schrödinger equation for the harmonic oscillator. The solution for all r can be expressed as where N n is a normalization constant. Moreover, the spectrum is expressed as (for EΩ > 0) The eigenvalue of energy (2.25) obtained in Ref. [1] is not similar to our result (12) due to it having been obtained from a wrong differential equation.
As a second example, let us consider an attractive Coulomb potential V (r ) = − a r . By introducing the Coulomb potential into Eq. (6), and using ψ(r ) = ψ + ψ − and σ 3 ψ s (r ) = sψ s (r ) with s = ±1, we get where The solution for (13), with C necessarily real and positive, is the solution of the Schrödinger equation for the threedimensional harmonic oscillator plus a Cornell potential [3][4][5][6]. By setting and by introducing the new variable and parameters one finds that the solution for all r can be expressed as a solution of the biconfluent Heun differential equation [5][6][7][8][9][10][11] x with Θ = 1 2 [δ + ρ (ω + 1)] and It is well known that the biconfluent Heun equation has a regular singularity at x = 0 and an irregular singularity at x = ∞ [4]. The regular solution at the origin is where Γ (z) is the gamma function, A 0 = 1, A 1 = Θ and the remaining coefficients for ρ = 0 satisfy the recurrence relation where Δ = τ − ω − 2. The series is convergent and tends to exp r as x → ∞. This asymptotic behavior perverts the normalizability of the solution (18), because ψ(r ) ∝ exp and The condition (28) furnishes a polynomial of degree n + 1 in δ; there are at most n + 1 suitable values of δ. At this stage, it is worth to mention that the energy of the system is obtained using both conditions (27) and (28).
The problem does not end here; it is necessary to analyze the condition (28). For n = 0, the condition (28) becomes A 1 = Θ = 0 and results in an algebraic equation of degree one in δ, Substituting (20), (21) and (24) into (31), we obtain where m = 2 m α + s 2 + 2 m α − s 2 + 1. Substituting (32) into (29) for n = 0, we have Equation (33) represents the energy eigenvalue for n = 0. For n = 1, the condition (28) becomes A 2 = 0 and results in an algebraic equation of degree two in δ. For n ≥ 2 the algebraic equations are cumbersome. In this comment, we will only consider the solution for n = 0 for simplicity. In summary, we studied the Dirac equation in the presence of scalar and vector potentials in a class of flat Gödel-type space-times called Som-Raychaudhuri space-times. We calculated the correct second-order differential equation for this system. As in Ref. [1], we have considered two cases: (1) V (r ) = 0 and (2) the Coulomb potential. For the first case, V (r ) = 0, the problem was mapped into a Schrödinger-like equation with the harmonic oscillator potential. The correct energy spectrum for this case was obtained. For the second case, we considered an attractive Coulomb potential. In this case, the problem was mapped into a biconfluent Heun differential equation and appropriately using the quantization conditions (27) and (28), we found the correct energy spectrum for n = 0. Finally, we showed that the results obtained in Ref. [1] are incorrect, due to them having been obtained from a wrong differential equation.

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