Abstract
We show that from an appropriate manipulation of the biconfluent Heun differential equation one can obtain the correct expression for the energy eigenvalues for the Klein–Gordon equation without potential in the background of Som–Raychaudhuri space-time with a cosmic string as a particular case (\(k_{L}=0\)) of Vitória et al. (Eur Phys J C 78(1):44, 2018), in opposition what was stated in a recent paper published in this journal (Ahmed in Eur Phys J C 79(8):682, 2019).
Avoid common mistakes on your manuscript.
In a recent paper in this journal, Ahmed [1] has pointed out an incorrect expression in [2]. The author has solved the Klein–Gordon equation without interaction in the Som–Raychaudhuri space-time with the cosmic string using the Nikiforov–Uvarov method and obtained the energy eigenvalues and the corresponding eigenfunctions of the system. The problem solved in [1] is a special case associated to \(k_{L}=0\) in Ref. [2]. Ahmed has claimed that to obtain the correct result, one should start from Eq. (4) but not from the condition (12), which is obtained from the biconfluent Heun equation (10). The purpose of this comment is to show that following an appropriate manipulation of the biconfluent Heun equation, the correct energy eigenvalues for \(k_{L}=0\) can be obtained as a particular case of [2], as opposed to what was advertised in Ref. [1].
The Klein–Gordon equation in the Som–Raychaudhuri space-time with a linear scalar potential is given by
Considering the solution in the form
and by introducing the new variable and parameters
one finds that Eq. (1) becomes
The solution for (8) can be expressed as
where H(x) can be expressed as a solution of the biconfluent Heun differential equation [3,4,5,6]
with
and
The differential equation (10) has a regular singularity at \(x=0\) and an irregular singularity at \(x=\infty \). The regular solution at the origin is given by
where \(\varGamma (z)\) is the gamma function, \(A_{0}=1\), \(A_{1}=\sigma \) and the remaining coefficients for \(\theta \ne 0\) satisfy the recurrence relation,
where \(\varDelta =\tau -\frac{2|l|}{\alpha }-2\). From the recurrence (14), the solution H becomes a polynomial of degree n if and only if \(\varDelta =2n\) (\(n=0,1,\ldots \)) and \(A_{n+1}=0\).
On the other hand, if \(\theta =0\), the solution H becomes a polynomial of degree n if and only if
In fact, when \(\varDelta =4n\), one has [4]
where \(L^{(\delta )}_{n}(x)\) denotes the generalized Laguerre polynomial, a polynomial of degree n with n distinct positive zeros in the range \([0,\infty )\). This result is very important for obtaining the correct energy eigenvalues for the particular case \(k_{L}=0\) (\(\theta =0\)).
The especial case \(k_{L}=0\) (\(\theta =0\)) was studied in [1]. The author of [1] concluded that the correct expression of the energy cannot be obtained as a particular case from the biconfluent Heun differential equation, but this statement is false. Considering \(k_{L}=0\) (\(\theta =0\)) and using the correct condition (15), we obtain
Substituting (4), (5) and (7) into (17), we obtain the spectrum for \(k_{L}=0\) as (for \(E\varOmega >0\))
and the solution becomes
where \(N_{n}\) is a normalization constant. Equation (18) is the same as the result obtained in Ref. [7].
In summary, we showed that the correct expression for the energy eigenvalues for the case \(k_{L}=0\) of Ref. [2] can be obtained from an appropriate manipulation of the biconfluent Heun differential equation, in opposition to what was concluded in [1]. The results obtained in this comment are consistent with those found in [7].
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental data has been listed.]
References
F. Ahmed, Eur. Phys. J. C 79(8), 682 (2019). https://doi.org/10.1140/epjc/s10052-019-7196-3
R.L.L. Vitória, C. Furtado, K. Bakke, Eur. Phys. J. C 78(1), 44 (2018). https://doi.org/10.1140/epjc/s10052-018-5524-7
B. Leaute, G. Marcilhacy, J. Phys. A: Math. and Gen. 19(17), 3527 (1986). http://stacks.iop.org/0305-4470/19/i=17/a=017
A. Ronveaux, Heun’s differential equations (Oxford University Press, Oxford, 1986)
L.B. Castro, Phys. Rev. C 86, 052201 (2012). https://doi.org/10.1103/PhysRevC.86.052201
F.A.C. Neto, L.B. Castro, Eur. Phys. J. C 78(6), 494 (2018). https://doi.org/10.1140/epjc/s10052-018-5847-4
J. Carvalho, A.M. de Carvalho, C. Furtado, Eur. Phys. J. C 74(6), 2935 (2014). https://doi.org/10.1140/epjc/s10052-014-2935-y
Acknowledgements
This work was supported in part by means of funds provided by CNPq, Brazil, Grant No. 307932/2017-6 (PQ) and No. 422755/2018-4 (UNIVERSAL), São Paulo Research Foundation (FAPESP), Grant No. 2018/20577-4, FAPEMA, Brazil, Grant No. UNIVERSAL-01220/18 and CAPES, Brazil.
Author information
Authors and Affiliations
Corresponding author
Additional information
This comment refers to the article available at https://doi.org/10.1140/epjc/s10052-019-7196-3.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3
About this article
Cite this article
Neto, F.A.C., Soares, C.C. & Castro, L.B. Comment on “Comment on Linear confinement of a scalar particle in a Gödel-type space-time”. Eur. Phys. J. C 80, 53 (2020). https://doi.org/10.1140/epjc/s10052-019-7603-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-019-7603-9