# Comment on “Effects of cosmic-string framework on the thermodynamical properties of anharmonic oscillator using the ordinary statistics and the *q*-deformed superstatistics approaches”

## Abstract

We point out a misleading treatment in a recent paper published in this Journal (Sobhani et al., Eur Phys J C 78:106, 2018) regarding solutions for the Schrödinger equation with a anharmonic oscillator potential embedded in the background of a cosmic string mapped into biconfluent Heun equation. This fact jeopardizes the thermodynamical properties calculated in this system.

In a recent paper in this Journal, Sobhami et. al. [1] have studied the thermodynamical properties of the anharmonic oscillator within cosmic-string framework using ordinary statistic and the *q*-deformed superstatistics approaches. To achieve their goal, the authors need to calculate the wave function and the energy spectrum, which have been obtained from the Schrödinger equation within a cosmic-string framework mapped into biconfluent Heun differential equation. It is worthwhile to mention that all results depend mainly on the energy spectrum of the system. The purpose of this comment is point to out a misleading treatment on the solution of the biconfluent Heun equation, this fact jeopardizes the results of [1].

*C*necessarily real and positive (\(c_{v}>0\)), is the solution of the Schrödinger equation for the three-dimensional harmonic oscillator plus a Cornell potential [2, 3, 4]. It is worthwhile to mention that Refs. [2, 3] present some erroneous calculations.

*f*(

*y*), i.e \(f(y)\propto \exp \left( \frac{\sqrt{c_{v}}}{4}y^{2}+\frac{b_{v}}{4\sqrt{c_{v}}}y \right) \) as \(y\rightarrow \infty \). Nevertheless, this trouble can be surpassed by considering a polynomial solution for \(H_{b}\left( 0,\beta ^{\prime },\gamma ^{\prime },\delta ^{\prime };x\right) \). In fact, \(H_{b}\left( 0,\beta ^{\prime },\gamma ^{\prime },\delta ^{\prime };x\right) \) presents polynomial solutions of degree

*n*if and only if two conditions are satisfied:

*n*, \(c_{v}\), \(a_{v}\),

*l*and \(\alpha \). The problem does not end here, it is necessary to analyze the second condition of quantization.

*n*. This is a peculiar behavior of the biconfluent Heun equation. Our results show that the expression for the energy in Ref. [1] is wrong, probably due to erroneous calculations in the manipulation of the biconfluent Heun equation. The correct quantization condition is obtained applying two conditions: the condition (17) is used to obtain constraints between the potential parameters and the condition (18) is used to obtain the expression of energy eigenvalues. It is worthwhile mention that condition of quantization is different for each value of

*n*.

In summary, we analyzed the solution for the Schrödinger equation with a anharmonic oscillator potential embedded in the background of a cosmic string. In this process, the problem is mapped into biconfluent Heun differential equation and using appropriately the conditions (17) and (18), we found the correct energy eigenvalues and constraints on the potential parameters. Also, we showed that there is no need for fixing the value of \(c_{v}=4\) for obtain a biconfluent Heun equation, in contrast to Ref. [1]. Additionally, the thermodynamical properties calculated in Ref. [1] depend of the energy spectrum relation, therefore our results jeopardize the main results of Ref. [1].

## Notes

### Acknowledgements

We acknowledge valuable comments from the anonymous referee. This work was supported in part by means of funds provided by CNPq, Brazil, Grant No. 307932/2017-6 (PQ).

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