Comment on “Effects of cosmicstring framework on the thermodynamical properties of anharmonic oscillator using the ordinary statistics and the qdeformed superstatistics approaches”
Eur. Phys. J. C
Comment on “Effects of cosmicstring framework on the thermodynamical properties of anharmonic oscillator using the ordinary statistics and the q deformed superstatistics approaches”
Francisco A. Cruz Neto 0
Luis B. Castro 0
0 Departamento de Física, Universidade Federal do Maranhão , Campus Universitário do Bacanga, São Luís, MA 65080805 , Brazil
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 cosmicstring framework using ordinary
statistic and the qdeformed 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 cosmicstring
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
].
The timeindependent Schrödinger equation with an
anharmonic oscillator potential embedded in the background
of a cosmic string is given by
d2Φ(ρ)
dρ2
1 dΦ(ρ)
+ ρ dρ
2
+ ε − kz −
av + l2α2 ρ2 − bvρ4 − cvρ6
Φ(ρ) = 0.
Redefining the wave function as Φ(ρ) = R√(ρρ) , one can
remove the first derivative and rewrite Eq. (1) as
The solution for the differential Eq. (3) with C necessarily
real and positive (cv > 0), is the solution of the Schrödinger
equation for the threedimensional harmonic oscillator plus
a Cornell potential [
2–4
]. It is worthwhile to mention that
Refs. [
2,3
] present some erroneous calculations.
Considering the solution in the form of [
3
]
f (y) = y1/2 exp
− √4cv y2 − 4 √bvcv y F (y),
(1)
and by introducing the new variable and parameters:
x =
β =
γ =
cv 1/4
4
4
cv
y ,
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
d2 R(ρ)
dρ2
1 2
ε + 4ρ2 − kz −
−bvρ4 − cvρ6
R(ρ) = 0.
av + l2α2 ρ2
Making use of the new variable y = ρ2 and redefining the
wave function as R(y) = f√4(yy) , the Eq. (2) becomes
A 1/4
κ + y − B y − C y2 + y2
f (y) = 0 ,
d y2
where
B = b4v ,
C = c4v .
2 2 ,
av + l α
ε − kz2 ,
with
1
Θ = 2 δ + β
where
,
one finds that the solution can be expressed as a solution of
the biconfluent Heun differential equation [
5–9
]
d2 F (x )
d x 2
δ = c4v 2 . (14)
This differential equation has a regular singularity at x = 0
and an irregular singularity at x = ∞. The regular solution
at the origin is given by
Hb 0, β , γ , δ ; x =
∞
j=0
1 A j j
x ,
Γ (1 + j ) j !
here Γ (z) is the gamma function, A0 = 1, A1 = Θ and
the remaining coefficients for β = 0 satisfy the recurrence
relation,
A j+2 = ( j + 1)β + Θ A j+1 − ( j + 1)2(Δ − 2 j ) A j ,
(16)
where Δ = γ − 2.
The series is convergent and tends to exp x 2 + β x as
x → ∞. It is true that the presence of exp x 2 + β x
in the asymptotic behavior of Hb 0, β , γ , δ ; x perverts
the normalizability of the solution f (y), i.e f (y) ∝
exp √4cv y2 + 4 √bvcv y as y → ∞. Nevertheless, this
trouble can be surpassed by considering a polynomial solution
for Hb 0, β , γ , δ ; x . In fact, Hb 0, β , γ , δ ; x presents
polynomial solutions of degree n if and only if two conditions
are satisfied:
Δ = 2n,
and
An+1 = 0 .
n = 0, 1, 2, . . .
Now, the condition (18) provides a polynomial of degree
n + 1 in δ and there are at most n + 1 suitable values of δ .
Therefore, the energy eigenvalues of the system are obtained
for the both conditions (17) and (18).
From the condition (17), one obtains
bv,n = ±
where
4
σ = 16 cv
1/2
128(n + 1) + σ ,
2 2 .
av + l α
(12)
(13)
(15)
(17)
(18)
(19)
(20)
This result shows a constraint that involves a specific value
for bv as a function of n, cv, av, l and α. The problem does
not end here, it is necessary to analyze the second condition
of quantization.
Now, we focus attention on the condition (18). For n = 0,
the condition A1 = Θ = 0 results in an algebraic equation
of degree one in δ :
δ + β = 0 ,
which furnishes the following expression for the energy
1/2
ε0 = kz2 + bv2,0 c4v . (21)
Note that this result presents two different classes of solutions
that depend on the sign of bv,0. If A ≷ 0 [Eq. (5)], that implies
ε ≷ kz2, then bv,0 ≷ 0, so we get
ε0,± = kz2 ± 21 c4v 1/4 √128 + σ . (22)
The expression (22) represents the energy eigenvalue for n =
0.
Now, let us consider the case for n = 1, which implies
that A2 = β + Θ Θ−2 = 0. In this case we obtain an
algebraic equation of degree two in δ :
2
δ + β
+ 2β δ + β
− 8 = 0 ,
which furnishes the energy eigenvalues
2
ε1,1 = kz +
and
2
ε1,2 = kz +
4
cv
4
cv
1/2
1/2
1
bv,1 + 2
1
bv,1 − 2
bv,1
2 + 128
bv,1
2 +128
cv 3/2
4
cv 3/2
4
(23)
.
(24)
For simplicity, we will only consider the case bv,1 > 0
[positive sign in (19)]. Therefore, the energy eigenvalues in this
case are given by
2
ε1,1,+ = kz +
and
2
ε1,1,+ = kz +
cv 1/4 √
4
1 √
σ + 256 + 2
σ + 384
(25)
cv 1/4 √
4
1 √
σ + 256 − 2 σ + 384 . (26)
For the case of n = 2, the condition A3 = 0 results in an
algebraic equation of degree three in δ . And the solutions, in
fact, turn out to be three values for the energy (considering
bv,3 > 0). For n ≥ 2 the algebraic equations are
cumbersome. With this result, we conclude that the energy can not be
labeled with 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 cv = 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
].
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. 307932/20176 (PQ).
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