effects of kaluza-klein theory and potential on a generalized ...kaluza-klein theory subject toa...

Research ArticleEffects of Kaluza-Klein Theory and Potential on a GeneralizedKlein-Gordon Oscillator in the Cosmic String Space-Time

Faizuddin Ahmed

Ajmal College of Arts and Science, Dhubri, 783324 Assam, India

Correspondence should be addressed to Faizuddin Ahmed; [email protected]

Received 19 April 2020; Revised 15 May 2020; Accepted 22 May 2020; Published 10 July 2020

Academic Editor: Shi-Hai Dong

Copyright © 2020 Faizuddin Ahmed. 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.

In this paper, we solve a generalized Klein-Gordon oscillator in the cosmic string space-time with a scalar potential of Cornell-typewithin the Kaluza-Klein theory and obtain the relativistic energy eigenvalues and eigenfunctions. We extend this analysis byreplacing the Cornell-type with Coulomb-type potential in the magnetic cosmic string space-time and analyze a relativisticanalogue of the Aharonov-Bohm effect for bound states.

1. Introduction

A unified formulation of Einstein’s theory of gravitation andtheory of electromagnetism in four-dimensional space-timewas first proposed by Kaluza [1] by assuming a pure gravita-tional theory in five-dimensional space-time. The so-calledcylinder condition was later explained by Klein when theextra dimension was compactified on a circle S1 with amicroscopic radius [2], where the spatial dimension becomesfive-dimensional. The idea behind introducing additionalspace-time dimensions has found application in quantumfield theory, for instance, in string theory [3]. There arestudies on Kaluza-Klein theory with torsion [4, 5], in theGrassmannian context [6–8], in Kähler fields [9], in the pres-ence of fermions [10–12], and in the Lorentz-symmetryviolation (LSV) [13–15]. Also, there are investigations inspace-times with a topological defect in the context ofKaluza-Klein theory, for example, the magnetic cosmic string[16] (see also [17]), and magnetic chiral cosmic string [18] infive-dimensions.

Aharonov-Bohm effect [19–21] is a quantum mechanicalphenomena that has been investigated in several branches ofphysics, such as in, graphene [22], Newtonian theory [23],bound states of massive fermions [24], scattering of dislo-cated wave-fronts [25], torsion effect on a relativisticposition-dependent mass system [26, 27], and nonminimalLorentz-violating coupling [28]. In addition, Aharonov-

Bohm effect has been investigated in the context of Kaluza-Klein theory by several authors [18, 29–34], and the geomet-ric quantum phase in graphene [35]. It is well-known in con-densed matter [36–40] and in the relativistic quantumsystems [41, 42] that when there exists dependence of theenergy eigenvalues on geometric quantum phase [21], then,persistent current arises in the systems. The studies of persis-tent currents have explored systems that deal with the Berryphase [43, 44], the Aharonov-Anandan quantum phase [45,46], and the Aharonov-Casher geometric quantum phase[47–50]. Investigation of magnetization and persistent cur-rents of mass-less Dirac Fermions confined in a quantumdot in a graphene layer with topological defects was studiedin [51].

Klein-Gordon oscillator theory [52, 53] was inspired bythe Dirac oscillator [54]. This oscillator field is used to studythe spectral distribution of energy eigenvalues and eigenfunc-tions in 1 − d version of Minkowski space-times [55]. Klein-Gordon oscillator was studied by several authors, such as inthe cosmic string space-time with external fields [56], withCoulomb-type potential by two ways: (i) modifying the massterm m⟶m + SðrÞ [57], and (ii) via the minimal coupling[58] in addition to a linear scalar potential, in the back-ground space-time generated by a cosmic string [59], inthe Gödel-type space-times under the influence of gravita-tional fields produced by topological defects [60], in theSom-Raychaudhuri space-time with a disclination parameter

HindawiAdvances in High Energy PhysicsVolume 2020, Article ID 8107025, 15 pageshttps://doi.org/10.1155/2020/8107025

https://orcid.org/0000-0003-2196-9622

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https://doi.org/10.1155/2020/8107025

[61], in noncommutative (NC) phase space [62], in ð1 + 2Þ-dimensional Gürses space-time background [63], and inð1 + 2Þ-dimensional Gürses space-time background subjectto a Coulomb-type potential [64]. The relativistic quantumeffects on oscillator field with a linear confining potentialwere investigated in [65].

We consider a generalization of the oscillator asdescribed in Refs. [34, 64] for the Klein-Gordon. This gener-alization is introduced through a generalized momentumoperator where the radial coordinate r is replaced by a gen-eral function f ðrÞ. To author’s best knowledge, such a newcoupling was first introduced by Bakke et al. in Ref. [41]and led to a generalization of the Tan-Inkson model of atwo-dimensional quantum ring for systems whose energylevels depend on the coupling’s control parameters. Basedon this, a generalized Dirac oscillator in the cosmic stringspace-time was studied by Deng et al. in Ref. [66] where thefour-momentum pμ is replaced with its alternative pμ +mω

βf μðxμÞ. In the literature, f μðxμÞ has chosen similar to poten-tials encountered in quantum mechanics (Cornell-type,exponential-type, singular, Morse-type, Yukawa-like etc.). Ageneralized Dirac oscillator in ð2 + 1Þ-dimensional worldwas studied in [67]. Very recently, the generalized K-G oscil-lator in the cosmic string space-time in [68] and noninertialeffects on a generalized DKP oscillator in the cosmic stringspace-time in [69] were studied.

The relativistic quantum dynamics of a scalar particle ofmass m with a scalar potential SðrÞ [70, 71] is described bythe following Klein-Gordon equation:

1ffiffiffiffiffiffi−gp ∂μffiffiffiffiffiffi−g

pgμν∂νð Þ − m + Sð Þ2

� �Ψ = 0, ð1Þ

with g is the determinant of metric tensor with gμν itsinverse. To couple Klein-Gordon field with oscillator [52,53], following change in the momentum operator is consid-ered as in [56, 72]:

p!⟶ p

! + imω r!, ð2Þ

where ω is the oscillatory frequency of the particle and r! = r r̂where r being distance from the particle to the string. Togeneralized the Klein-Gordon oscillator, we adopted theidea considered in Refs. [34, 64, 66, 68, 69] by replacingr⟶ f ðrÞ as

Xμ = 0, f rð Þ, 0, 0, 0ð Þ: ð3Þ

So we can write p!⟶ p

! + i mω f ðrÞ r̂, and we have

p2 ⟶ ðp! + i mω f ðrÞ r!Þðp! − i mω f ðrÞ r!Þ. Therefore, thegeneralized Klein-Gordon oscillator equation:

1ffiffiffiffiffiffi−gp ∂μ +mωXμ

� � ffiffiffiffiffiffi−g

pgμν ∂ν −mωXνð Þf g − m + Sð Þ2

� �Ψ = 0,

ð4Þ

where Xμ is given by Eq. (3).

Various potentials have been used to investigate thebound state solutions to the relativistic wave-equations.Among them, much attention has given on Coulomb-typepotential. This kind of potential has widely used to study var-ious physical phenomena, such as in, the propagation ofgravitational waves [73], the confinement of quark models[74], molecular models [75], position-dependent mass sys-tems [76–78], and relativistic quantum mechanics [57–59].The Coulomb-type potential is given by

S rð Þ = ηcr: ð5Þ

where ηc is the Coulombic confining parameter.Another potential that we are interested here is the

Cornell-type potential. The Cornell potential, which consistsof a linear potential plus a Coulomb potential, is a particularcase of the quark-antiquark interaction, one more harmonictype term [79]. The Coulomb potential is responsible by theinteraction at small distances, and the linear potential leadsto the confinement. Recently, the Cornell potential has beenstudied in the ground state of three quarks [80]. However,this type of potential is worked on spherical symmetry; incylindrical symmetry, which is our case, this type of potentialis known as Cornell-type potential [60]. This type of interac-tion has been studied in [60, 65, 70, 81]. Given this, let usconsider this type of potential

S rð Þ = ηL r +ηcr, ð6Þ

where ηL, ηc are the confining potential parameters.The aim of the present work is to analyze a relativistic

analogue of the Aharonov-Bohm effect for bound states[19–21] for a relativistic scalar particle with potential in thecontext of Kaluza-Klein theory. First, we study a relativisticscalar particle by solving the generalized Klein-Gordon oscil-lator with a Cornell-type potential in the five-dimensionalcosmic string space-time. Secondly, by using the Kaluza-Klein theory [1–3], a magnetic flux through the line-element of the cosmic string space-time is introduced andthus wrote the generalized Klein-Gordon oscillator in thefive-dimensional space-time. In the later case, a Coulomb-type potential by modifying the mass term m⟶m + SðrÞis introduced which was not studied earlier. Then, we showthat the relativistic bound states solutions can be achieved,where the relativistic energy eigenvalues depend on the geo-metric quantum phase [21]. Due to this dependence of therelativistic energy eigenvalue on the geometric quantumphase, we calculate the persistent currents [36, 37] that arisein the relativistic system.

This paper comprises as follows: In section 2, we study ageneralized Klein-Gordon oscillator in the cosmic stringbackground within the Kaluza-Klein theory with a Cornell-type scalar potential; in section 3, a generalized Klein-Gordon oscillator in the magnetic cosmic string in theKaluza-Klein theory subject to a Coulomb-type scalar poten-tial and obtain the energy eigenvalues and eigenfunctions,and the conclusion one in section 4.

2 Advances in High Energy Physics

2. Generalized Klein-Gordon Oscillator inCosmic String Space-Time with a Cornell-Type Potential in Kaluza-Klein Theory

The purpose of this section is to study the Klein-Gordonequation in cosmic string space-time with the use ofKaluza-Klein theory with interactions. The first study of thetopological defects within the Kaluza-Klein theory was car-ried out in [16]. The metric corresponding to this geometrycan be written as,

ds2 = −dt2 + dr2 + α2 r2 dϕ2 + dz2 + dx2, ð7Þ

where t is the time coordinate, x is the coordinate associatedwith the fifth additional dimension, and ðr, ϕ, zÞ are cylindri-cal coordinates. These coordinates assume the ranges −∞ <ðt, zÞ <∞, 0 ≤ r <∞, 0 ≤ ϕ ≤ 2π, 0 < x < 2π a, where a isthe radius of the compact dimension x. The α parametercharacterizing the cosmic string, and in terms of mass densityμ given by α = 1 − 4 μ [82]. The cosmology and gravitationimpose limits to the range of the α parameter which isrestricted to α < 1 [82].

By considering the line element (7) into the Eq. (4), weobtain the following differential equation:

−∂2

∂t2+ 1r

∂∂r

+mω f rð Þ� �

r∂∂r

−mω r f rð Þ� �"

+ 1α2 r2

∂2

∂ϕ2+ ∂2

∂z2+ ∂∂x2

− m + S rð Þð Þ2#Ψ t, r, ϕ, z, xð Þ = 0

−∂2

∂t2+ ∂2

∂r2+ 1r

∂∂r

−mω f ′ rð Þ + f rð Þr

� �−m2 ω2 f 2 rð Þ

"

+ 1α2 r2

∂2

∂ϕ2+ ∂2

∂z2+ ∂∂x2

− m + S rð Þð Þ2#Ψ t, r, ϕ, z, xð Þ = 0:

ð8Þ

Since the metric is independent of t, ϕ, z, x. One canchoose the following ansatz for the function Ψ

Ψ t, r, ϕ, z, xð Þ = ei −E t+l ϕ+k z+q xð Þ ψ rð Þ, ð9Þ

where E is the total energy, l = 0, ±1,±2::, and k, q areconstants.

Substituting the above ansatz into the Eq. (8), we get thefollowing equation for ψðrÞ:

d2

dr2+ 1r

ddr

+ E2 − k2 − q2 −l2

α2 r2−mω f ′ rð Þ + f rð Þ

r

� �"

−m2 ω2 f 2 rð Þ − m + S rð Þð Þ2#ψ rð Þ = 0:

ð10Þ

We choose the function f ðrÞ a Cornell-type given by [34,64, 66, 69]

f rð Þ = a r + br, a, b > 0: ð11Þ

Substituting the function (11) and Cornell potential (6)into the Eq. (9), we obtain the following equation:

d2

dr2+ 1r

ddr

+ λ −Ω2 r2 −j2

r2−2m ηc

r− 2m ηL r

" #ψ rð Þ = 0,

ð12Þ

where

λ = E2 − k2 − q2 −m2 − 2mω a − 2m2 ω2 a b − 2 ηL ηc,

Ω =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 ω2 a2 + η2L

q,

j =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

α2+m2 ω2 b2 + η2c

r:

ð13Þ

Transforming ρ =ffiffiffiffiΩ

pr into the Eq. (12), we get

d2

dρ2+ 1ρ

ddρ

+ ζ − ρ2 −j2

ρ2−η

ρ− θ ρ

" #ψ ρð Þ = 0, ð14Þ

where

ζ = λ

Ω,

η = 2m ηcffiffiffiffiΩ

p ,

θ = 2m ηLΩ3/2 :

ð15Þ

Let us impose that ψðρÞ⟶ 0 when ρ⟶ 0 and ρ⟶∞. Suppose the possible solution to the Eq. (14) is

ψ ρð Þ = ρj e− 1/2ð Þ ρ+θð Þ ρ H ρð Þ: ð16Þ

Substituting the solution Eq. (16) into the Eq. (14), weobtain

H″ ρð Þ + γ

ρ− θ − 2 ρ

� �H ′ ρð Þ + −

β

ρ+Θ

� �H ρð Þ = 0, ð17Þ

where

γ = 1 + 2 j,

Θ = ζ + θ2

4 − 2 1 + jð Þ,

β = η + θ

2 1 + 2 jð Þ:

ð18Þ

3Advances in High Energy Physics

Eq. (17) is the biconfluent Heun’s differential equation[26, 27, 29–34, 58–61, 64, 65, 70, 83–87], and HðρÞ is theHeun polynomials.

The above Eq. (17) can be solved by the Frobeniusmethod. We consider the power series solution around theorigin [88]

H ρð Þ = 〠∞

i=0ci ρ

i ð19Þ

Substituting the above power series solution into theEq. (17), we obtain the following recurrence relation forthe coefficients:

cn+2 =1

n + 2ð Þ n + 2 + 2 jð Þ β + θ n + 1ð Þf g cn+1 − Θ − 2 nð Þ cn½ �:

ð20Þ

And the various coefficients are

c1 =η

γ−θ

2

� �c0, c2 =

14 1 + jð Þ β + θð Þ c1 −Θ c0½ �: ð21Þ

The quantum theory requires that the wave function Ψmust be normalized. The bound state solutions ψðρÞ canbe obtained because there is no divergence of the wavefunction at ρ⟶ 0 and ρ⟶∞. Since we have writtenthe function HðρÞ as a power series expansion aroundthe origin in Eq. (19). Thereby, bound state solutionscan be achieved by imposing that the power series expan-sion (19) becomes a polynomial of degree n. Through therecurrence relation (20), we can see that the power seriesexpansion (19) becomes a polynomial of degree n byimposing two conditions [26, 27, 29–34, 58–61, 64, 65,70, 83–85]:

Θ = 2 n n = 1, 2,⋯ð Þ, cn+1 = 0 ð22Þ

By analyzing the condition Θ = 2 n, we get the expres-sion of the energy eigenvalues En,l:

λ

Ω+ θ2

4 − 2 1 + jð Þ = 2 n⇒ E2n,l = k2 + q2 +m2 + 2Ω

� n + 1 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

α2+m2 ω2 b2 + η2c

r !

+ 2m2 ω2 a b + 2mω a + 2 ηL ηc −m2 η2LΩ2 :

ð23Þ

We plot graphs of the above energy eigenvalues w. r. t. dif-ferent parameters. In Figure 1, the energy eigenvalues E1,1against the parameter ηc. In Figure 2, the energy eigenvaluesE1,1 against the parameter ηL. In Figure 3, the energy eigen-values E1,1 against the parameter M. In Figure 4, the energyeigenvalues E1,1 against the parameter ω. In Figure 5, theenergy eigenvalues E1,1 against the parameter Ω.

Now, we impose an additional condition cn+1 = 0 to findthe individual energy levels and corresponding wave functions

0 2 4 6 8 10

0

1

2

3E 1,1

(𝜂c)

𝜂c

4

5

6

7

Figure 1: n = l = k =M = q = a = b = ηL = 1, α = 0:5, ω = 0:5:

E 1,1

(𝜂L)

𝜂L

0 2 4 6 8 10

0

2

4

6

8

10

Figure 2: n = l = k =M = q = a = b = ηc = 1, α = 0:5, ω = 0:5

0 2 4 6M

8 10

0

5

10

15

E 1,1

(M)

Figure 3: n = l = k = q = a = b = ηc = ηL = 1, α = 0:5, ω = 0:5


one by one as done in [89, 90]. As an example, for n = 1, wehave Θ = 2 and c2 = 0 which implies

c1 =2

β + θc0 ⇒

η

1 + 2 j −θ

2

� �= 2β + θ

Ω31,l −

η2

2 1 + 2 jð ÞΩ21,l

− η θ1 + j1 + 2 j

� �Ω1,l −

θ2

8 3 + 2 jð Þ = 0

ð24Þ

a constraint on the parameter Ω1,l. The relation given inEq. (24) gives the possible values of the parameter Ω1,l thatpermit us to construct first degree polynomial to HðxÞ forn = 1. Note that its values changes for each quantum numbern and l, so we have labeled Ω⟶Ωn,l. Besides, since thisparameter is determined by the frequency; hence, the fre-quency ω1,l is so adjusted that the Eq. (24) can be satisfied,where we have simplified our notation by labeling:

ω1,l =1ma

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩ2

1,l − η2L

q: ð25Þ

It is noteworthy that a third-degree algebraic Eq. (24)has at least one real solution, and it is exactly this solutionthat gives us the allowed values of the frequency for the low-est state of the system, which we do not write because itsexpression is very long. We can note, from Eq. (24), thatthe possible values of the frequency depend on the quantumnumbers and the potential parameter. In addition, for eachrelativistic energy level, we have a different relation of themagnetic field associated to the Cornell-type potential andquantum numbers of the system fl, ng. For this reason, wehave labeled the parameters Ω and ω in Eqs. (24) and (25).

Therefore, the ground state energy level and correspond-ing wave-function for n = 1 are given by

E21,l = k2 + q2 +m2 + 2Ω1,l 2 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

α2+m2 ω2 b2 + η2c

r !

+ 2m2 ω21,l a b + 2mω1,l a + 2 ηL ηc −

m2 η2LΩ2

1,l,

ψ1,l = ρffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2/α2+m2 ω2

1,l b2+η2c

pe− 1/2ð Þ 2m ηL/Ω3/2

1,l +ρð Þ ρ c0 + c1 ρð Þ,ð26Þ

where

c1 =1

Ω1/21,l

2m ηc

1 + 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2/α2� �

+m2 ω21,l b

2 + η2c

q −m ηLΩ1,l

264

375 c0:

ð27Þ

Then, by substituting the real solution of Eq. (25) intothe Eqs. (26)–(27), it is possible to obtain the allowed valuesof the relativistic energy for the radial mode n = 1 of a posi-tion dependent mass system. We can see that the lowestenergy state defined by the real solution of the algebraicequation given in Eq. (25) plus the expression given in Eq.(26) is defined by the radial mode n = 1, instead of n = 0. Thiseffect arises due to the presence of the Cornell-type potentialin the system.

For α⟶ 1, the relativistic energy eigenvalue (24)becomes

E2n,l = k2 + q2 +m2 + 2Ω n + 1 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2 +m2 ω2 b2 + η2c

q� �

+ 2m2 ω2 a b + 2mω a + 2 ηL ηc −m2 η2LΩ2 :

ð28Þ

Eq. (28) is the relativistic energy eigenvalue of scalar par-ticles via the generalized Klein-Gordon oscillator subject to aCornell-type potential in the Minkowski space-time in theKaluza-Klein theory.

We discuss bellow a very special case of the above relativ-istic system.

0 1 2 3 4 5

0

2

4

6

8

10

12

𝜔

E 1,1

(𝜔)

Figure 4: n = l = k = q = a = b = ηc = ηL =M = 1, α = 0:5

0 2 4 6 8 10

0

2

4

6

8

E 1,1

(𝛺)

𝛺

Figure 5: n = l = k = q = a = b = ηc = ηL =M = 1, α = 0:5, ω = 0:5


Case A. Considering ηL = 0, that is, only Coulomb-typepotential SðrÞ = ηc/r.

We want to investigate the effect of Coulomb-type poten-tial on a scalar particle in the background of cosmic stringspace-time in the Kaluza-Klein theory. In that case, the radialwave-equation Eq. (12) becomes

d2

dr2+ 1r

ddr

+ λ0 −m2 ω2 a2 r2 −j2

r2−2m ηcr

" #ψ rð Þ = 0,

ð29Þ

where

λ0 = E2 − k2 − q2 −m2 − 2mω a − 2m2 ω2 a b ð30Þ

Transforming ρ = ffiffiffiffiffiffiffiffiffiffiffimω a


d2

dρ2+ 1ρ

ddρ

+ λ0mω a

− ρ2 −j2

ρ2−

2m ηcffiffiffiffiffiffiffiffiffiffiffimω a

p 1ρ

" #ψ ρð Þ = 0:

ð31Þ

Suppose the possible solution to Eq. (31) is

ψ ρð Þ = ρj E− ρ2/2ð ÞH ρð Þ: ð32Þ


H″ ρð Þ + 1 + 2 jρ

− 2 ρ� �

H ′ ρð Þ + −~η

ρ+ λ0mω a

− 2 1 + jð Þ� �

H ρð Þ,

ð33Þ

where ~η = 2m ηc/ffiffiffiffiffiffiffiffiffiffiffimω a

p. Eq. (33) is the Heun’s differential

equation [26, 27, 29–34, 58–61, 64, 65, 70, 83–87] withHðρÞ is the Heun polynomial.

Substituting the power series solution Eq. (19) into theEq. (33), we obtain the following recurrence relation for coef-ficients

cn+2 =1

n + 2ð Þ n + 2 + 2 jð Þ ~η cn+1 −λ0

mω a− 2 1 + jð Þ − 2 n

� �cn

� �ð34Þ

The power series solution becomes a polynomial ofdegree n provided [26, 27, 29–34, 58–61, 64, 65, 70, 83–85]

λ0mω a

− 2 1 + jð Þ = 2 n n = 1, 2,⋯ð Þcn+1 = 0: ð35Þ

Using the first condition, one will get the followingenergy eigenvalues of the relativistic system:

En,l = ± k2 + q2 +m2 + 2mω a n + 2 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

α2+m2 ω2 b2 + η2c

r !(

+ 2m2 ω2 a b

)1/2

:

ð36Þ

The ground state energy levels and corresponding wave-function for n = 1 are given by

E1,l = ± k2 + q2 +m2 + 2mω1,l a 3 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

α2+m2 ω2 b2 + η2c

r !(

+ 2m2 ω2 a b

)1/2

, ψ1,l ρð Þ

= ρffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2/α2+m2 ω2

1,l b2+η2c

pe− ρ2/2ð Þ c0 + c1 ρð Þ,

ð37Þ

where

c1 =2m ηcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimω1,l a

p 1 + 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2/α2� �

+m2 ω21,l b

2 + η2c

q

= 21 + 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2/α2� �

+m2 ω21,l b

2 + η2c

q0B@

1CA

1/2

c0, ω1,l =2m η2c

a 1 + 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2/α2� �

+m2 ω21,l b

2 + η2c

q :

ð38Þ

a constraint on the frequency parameter ω1,l.

Case B. We consider another case corresponds to a⟶ 0,b⟶ 0 and ηL = 0, that is, a scalar quantum particle inthe cosmic string background subject to a Coulomb-typescalar potential within the Kaluza-Klein theory. In thatcase, from Eq. (12) we obtain the following equation:

ψ″ rð Þ + 1rψ′ rð Þ + ~λ −

~j2

r2−2m ηcr

" #ψ rð Þ = 0: ð39Þ

Eq. (39) can be written as

ψ″ rð Þ + 1rψ′ rð Þ + 1

r2−ξ1 r

2 + ξ2 r − ξ3� �

ψ rð Þ = 0, ð40Þ

where

ξ1 = −~λ = − E2 − k2 − q2 −m2� �,

ξ2 = −2m ηc,

ξ3 =~j2 = l2

α2+ η2c :

ð41Þ


Comparing the Eq (40) with Eq. (1) in the Appendix,we get

α1 = 1,α2 = 0,α3 = 0,α4 = 0,α5 = 0,α6 = ξ1,α7 = −ξ2,α8 = ξ3,α9 = ξ1,

α10 = 1 + 2ffiffiffiffiξ3

p,

α11 = 2ffiffiffiffiξ1

p,

α12 =ffiffiffiffiξ3

p,

α13 = −ffiffiffiffiξ1

p:

ð42Þ

The energy eigenvalues using Eqs. (41)–(42) into theEq. (8) in the Appendix is given by

En,l = ±m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − η2c

n +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2/α2� �

+ η2c

q+ 1/2

2 + k2

m2 + q2

m2

vuuut ,

ð43Þ

where n = 0, 1, 2, :: is the quantum number associated withthe radial modes, l = 0, ±1,±2,: are the quantum numberassociated with the angular momentum operator, k and qare arbitrary constants. Eq. (43) corresponds to the relativ-istic energy eigenvalues of a free-scalar particle subject to aCoulomb-type scalar potential in the background of cos-mic string within the Kaluza-Klein theory.

The corresponding radial wave-function is given by

ψn,l rð Þ = ∣N∣ r~j/2 e− r/2ð Þ L~jð Þn rð Þ

= ∣N∣ r1/2ffiffiffiffiffiffiffiffiffiffiffiffil2/α2+η2c

pe− r/2ð Þ L

ffiffiffiffiffiffiffiffiffiffiffiffil2/α2+η2c

p� �n rð Þ:

ð44Þ

Here, ∣N ∣ is the normalization constant and

Lðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl2/α2Þ+η2c

pÞ

n ðrÞ is the generalized Laguerre polynomial.For α⟶ 1, the relativistic energy eigenvalues Eq. (43)

becomes

En,l = ±mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − η2c

n +ffiffiffiffiffiffiffiffiffiffiffiffiffil2 + η2c

q+ 1/2

� �2 + k2

m2 + q2

m2

vuuut : ð45Þ

Eq. (45) corresponds to the relativistic energy eigen-value of a scalar particle subject to a Coulomb-type scalarpotential in the Minkowski space-time within the Kaluza-Klein theory.

3. Generalized Klein-Gordon Oscillator in theMagnetic Cosmic String with a Coulomb-Type Potential in Kaluza-Klein Theory

Let us consider the quantum dynamics of a particle movingin the magnetic cosmic string background. In the Kaluza-Klein theory [1, 2, 18], the corresponding metrics withAharonov-Bohm magnetic flux Φ passing along the symme-try axis of the string assumes the following form

ds2 = −dt2 + dr2 + α2 r2 dϕ2 + dz2 + dx + Φ

2π dϕ� �2

ð46Þ

with cylindrical coordinates are used. The quantum dynam-ics is described by the Eq. (4) with the following change inthe inverse matrix tensor gμν,

gμν =

−1 0 0 0 0 0 1 0 0 0

0 0 1

α2 r20 −

Φ

2π α2 r2

0 0 0 1 0

0 0 −Φ

2πα2 r2 0 1 + Φ2

4π2 α2 r2

0BBBBBBBBBB@

1CCCCCCCCCCA: ð47Þ

By considering the line element (46) into the Eq. (4), weobtain the following differential equation:

−∂2t + ∂2r +1r∂r +

1α2 r2

∂ϕ −Φ

2π ∂x

� �2+ ∂2z + ∂2x

"


� �−m2 ω2 f 2 rð Þ − m + S rð Þð Þ2

�Ψ = 0:

ð48Þ

Since the space-time is independent of t, ϕ, z, x,substituting the ansatz (9) into the Eq. (48), we get the fol-lowing equation:

ψ″ rð Þ + 1rψ′ rð Þ + E2 − k2 − q2 −

l2ef fr2


� �"

−m2 ω2 f 2 rð Þ − m + S rð Þð Þ2#ψ rð Þ = 0,

ð49Þ

where the effective angular quantum number

lef f =1α

l −qΦ2π

� �: ð50Þ


Substituting the function (11) into the Eq. (49) andusing Coulomb-type potential (5), the radial wave-equationbecomes

d2

dr2+ 1r

ddr

+ λ0 −m2 ω2 a2 r2 −χ2

r2−2m ηc

r

" #ψ rð Þ = 0,

ð51Þ

where

λ0 = E2 − k2 − q2 −m2 − 2mω a − 2m2 ω2 a b,

χ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2

α2+m2 ω2 b2 + η2c

r:

ð52Þ

Transforming ρ = ffiffiffiffiffiffiffiffiffiffiffimω a


d2

dρ2+ 1ρ

ddρ

+ λ0mω a

− ρ2 −χ2

ρ2−~η

ρ

" #ψ ρð Þ = 0, ð53Þ

where ~η = 2m ηc/ffiffiffiffiffiffiffiffiffiffiffimω a

p.

Suppose the possible solution to Eq. (53) is

ψ ρð Þ = ρχ e− ρ2/2ð ÞH ρð Þ ð54Þ


H″ ρð Þ + 1 + 2χρ

− 2 ρ� �

H ′ ρð Þ + −~η

ρ+ λ0mω a

− 2 1 + χð Þ� �

H ρð Þ:

ð55Þ

Eq. (55) is the second-order Heun’s differential equation[26, 27, 29–34, 58–61, 64, 65, 70, 83–87] with HðρÞ is theHeun polynomial.

Substituting the power series solution Eq. (19) into theEq. (55), we obtain the following recurrence relation for thecoefficients:

cn+2 =1

n + 2ð Þ n + 2 + 2χð Þ ~η cn+1 −λ0

mω a− 2 − 2χ − 2 n

� �cn

� �:

ð56Þ

The power series becomes a polynomial of degree n byimposing the following conditions [26, 27, 29–34, 58–61,64, 65, 70, 83–85]

cn+1 = 0, λ0mω a

− 2 − 2χ = 2 n n = 1, 2,⋯ð Þ ð57Þ

By analyzing the second condition, we get the followingenergy eigenvalues En,l:

E2n,l = k2 + q2 +m2 + 2mω a

� n + 2 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2

α2+m2 ω2 b2 + η2c

r !

+ 2m2 ω2 a b:

ð58Þ

Eq. (58) is the energy eigenvalues of a generalized Klein-Gordon oscillator in the magnetic cosmic string with aCoulomb-type scalar potential in the Kaluza-Klein theory.Observed that the relativistic energy eigenvalues Eq. (58)depend on the Aharonov-Bohm geometric quantum phase[21]. Thus, we have that En,lðΦ +Φ0Þ = En,l∓τðΦÞ where Φ0= ±ð2π/qÞ τwith τ = 0, 1, 2::. This dependence of the relativ-istic energy eigenvalue on the geometric quantum phase Φgives rise to a relativistic analogue of the Aharonov-Bohmeffect for bound states [18–21, 26, 29].

We plot graphs of the above energy eigenvalues w. r. t.different parameters. In Figure 6, the energy eigenvaluesE1,1 against the parameter ηc. In Figure 7, the energy eigen-values E1,1 against the parameter M. In Figure 8, the energyeigenvalues E1,1 against the parameter ω. In Figure 9, theenergy eigenvalues E1,1 against the parameter Φ.

The ground state energy levels and corresponding wave-function for n = 1 are given by

E21,l = k2 + q2 +m2 + 2mω1,l a

� 3 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2

α2+m2 ω2 b2 + η2c

r !

+ 2m2 ω21,l a b,

ψ1,l ρð Þ = ρffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil−qΦ/2πð Þ2/α2+m2 ω2

1,l b2+η2c

pe− ρ2/2ð Þ c0 + c1 ρð Þ,

ð59Þ

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

E 1,1 (𝜂c)

𝜂c

Figure 6: n = l = k = q = a = b =M = 1, α = 0:5, ω = 0:5, Φ = π


where

c1 =2m ηcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimω1,l a

p 1 + 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2/α2� �

+m2 ω21,l b

2 + η2c

q

= 21 + 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2/α2� �

+m2 ω21,l b

2 + η2c

q0B@

1CA

1/2

c0:

ω1,l =2m η2c

a 1 + 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2/α2� �

+m2 ω21,l b

2 + η2c

q ð60Þ

a constraint on the physical parameter ω1,l.Eq. (59) is the ground states energy eigenvalues and corre-

sponding eigenfunctions of a generalized Klein-Gordon oscil-lator in the presence of Coulomb-type scalar potential in amagnetic cosmic string space-time in the Kaluza-Klein theory.

For α⟶ 1, the energy eigenvalues (58) becomes

E2n,l = k2 +m2 + q2 + 2mω a

� n + 2 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil −

qΦ2π

� �2+m2 ω2 b2 + η2c

s0@

1A

+ 2m2 ω2 a b:

ð61Þ

Eq. (61) is the relativistic energy eigenvalue of the gener-alized Klein-Gordon oscillator field with a Coulomb-typescalar potential with a magnetic flux in the Kaluza-Klein the-ory. Observed that the relativistic energy eigenvalue Eq. (61)depend on the geometric quantum phase [21]. Thus, we havethat En,lðΦ +Φ0Þ = En,l∓τðΦÞ where Φ0 = ±ð2π/qÞ τ withτ = 0, 1, 2::. This dependence of the relativistic energy eigen-value on the geometric quantum phase gives rise to ananalogous effect to the Aharonov-Bohm effect for boundstates [18–21, 26, 29].

Case A. We discuss below a special case corresponds to b⟶ 0, a⟶ 0, that is, a scalar quantum particle in a mag-netic cosmic string background subject to a Coulomb-typescalar potential in the Kaluza-Klein theory. In that case, fromEq. (51), we obtain the following equation:

ψ″ rð Þ + 1rψ′ rð Þ + ~λ −

~χ2

r2−2m ηcr

� �ψ rð Þ = 0, ð62Þ

where

~λ = E2 − k2 − q2 −m2, ~χ0 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2

α2+ η2c

r: ð63Þ

The above Eq. (62) can be written as

ψ″ rð Þ + 1rψ′ rð Þ + 1

r2−ξ1 r

2 + ξ2 r − ξ3� �

ψ rð Þ = 0, ð64Þ

0 1 2 3 4 5

0

2

4

6

8

M

E 1,1

(M)

Figure 7: n = l = k = q = a = b = ηc = 1, α = 0:5, ω = 0:5, Φ = π

0 1 2 3 4 5

0

2

4

6

8

10

12

𝜔

E 1,1

(𝜔)

Figure 8: n = l = k = q = a = b = ηc =M = 1, α = 0:5, Φ = π

0 5 10 15 20 25 30

0

1

2

3

Ф

E 1,1

(Ф)

Figure 9: n = l = k = q = a = b = ηc =M = 1, α = 0:5, ω = 0:5


where

ξ1 = −~λ,ξ2 = −2m ηc,ξ3 = ~χ2

0:

ð65Þ

Following the similar technique as done earlier, we getthe following energy eigenvalues En,l:

En,l = ±mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − η2c

n +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/α2ð Þ l − qΦ/2πð Þ2 + η2c

q+ 1/2

� �2 + k2

m2 + q2

m2

vuuut ,

ð66Þ

where n = 0, 1, 2, :: is the quantum number associated withradial modes, l = 0, ±1, ±2,:⋯ are the quantum numberassociated with the angular momentum, k and q are con-stants. Eq. (66) corresponds to the relativistic energy levelsfor a free-scalar particle subject to Coulomb-type scalarpotential in the background of magnetic cosmic string in aKaluza-Klein theory.

The radial wave-function is given by

ψn,l rð Þ = ∣N∣ r~χ0/2 e− r/2ð Þ L~jð Þn rð Þ

= ∣N∣ r1/2,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil−qΦ/2πð Þ2/α2ð Þ+η2c

pe− r/2ð ÞL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil−qΦ/2πð Þ2/α2ð Þ+η2c

p� �n rð Þ:

ð67Þ

Here, ∣N ∣ is the normalization constant, and

Lðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiððl−qΦ/2πÞ2/α2Þ+η2c

pÞ

n ðrÞ is the generalized Laguerrepolynomial.

For α⟶ 1, the energy eigenvalues (66) becomes

En,l = ±mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − η2c

n +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2 + k2c

q+ 1/2

� �2 + k2

m2 + q2

m2

vuuut ,

ð68Þ

which is similar to the energy eigenvalue obtained in [30] (seeEq. (12) in [30]). Thus, we can see that the cosmic string αmodify the relativistic energy eigenvalue (66) in comparisonto those results obtained in [30].

Observe that the relativistic energy eigenvalues Eq. (66)depend on the cosmic string parameter α, the magneticquantum flux Φ, and potential parameter ηc. We can see thatEn,lðΦ +Φ0Þ = En,l∓τðΦÞ where Φ0 = ±ð2π/qÞ τ with τ = 0,1, ::. This dependence of the relativistic energy eigenvalueson the geometric quantum phase gives rise to a relativisticanalogue of the Aharonov-Bohm effect for bound states[18–21, 26, 29].

3.1. Persistent Currents of the Relativistic System. By follow-ing [36–38], the expression for the total persistent currentsis given by

I =〠n,lIn,l, ð69Þ

where

In,l = −∂En,l∂Φ

ð70Þ

is called the Byers-Yang relation (36).Therefore, the persistent current that arises in this relativ-

istic system using Eq. (58) is given by

where

∂χ∂Φ

= −q l − qΦ/2πð Þ

2 α2 πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2/α2� �

+m2 ω2 b2 + η2c

q : ð72Þ

Similarly, for the relativistic system discussed in Case Ain this section, this current using Eq. (66) is given by

In,l = −∂En,l∂Φ

= ∓mω a ∂χ/∂Φð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 + q2 +m2 + 2m2ω2ab + 2mωa n + 2 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2/α2� �

+m2ω2b2 + η2c

q r , ð71Þ

In,l = ± mq η2c l − qΦ/2πð Þ2π α2 n + 1/2 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2/α2� �

+ η2c

q 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil − qΦ/2πð Þ2/α2� �

+ η2c

q× 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − η2c / n +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/α2ð Þ l − qΦ/2πð Þ2 + η2c

q+ 1/2

� �2� �+ k2/m2� �

+ q2/m2ð Þs :

ð73Þ


For α⟶ 1, the persistent currents expression given byEq. (73) reduces to the result obtained in Ref. [30]. Thus,we can see that the presence of the cosmic string parametermodifies the persistent currents Eq. (73) in comparison tothose results in Ref. [30].

By introducing a magnetic flux through the line elementof the cosmic string space-time in five dimensions, we seethat the relativistic energy eigenvalue Eq. (58) depend onthe geometric quantum phase [21] which gives rise to a rel-ativistic analogue of the Aharonov-Bohm effect for boundstates [18–21, 26, 29]. Moreover, this dependence of the rel-ativistic energy eigenvalues on the geometric quantumphase has yielded persistent currents in this relativisticquantum system.

4. Conclusions

In Ref. [30], the Aharonov-Bohm effects for bound states of arelativistic scalar particle by solving the Klein-Gordon equa-tion subject to a Coulomb-type potential in the Minkowskispace-time within the Kaluza-Klein theory were studied.They obtained the relativistic bound states solutions and cal-culated the persistent currents. In Ref. [16], it is shown thatthe cosmic string space-time and the magnetic cosmic stringspace-time can have analogue in five dimensions. In Ref.[18], quantum mechanics of a scalar particle in the back-ground of a chiral cosmic string using the Kaluza-Klein the-ory was studied. They have shown that the wave functions,the phase shifts, and scattering amplitudes associated withthe particle depend on the global features of those space-times. This dependence represents the gravitational ana-logues of the well-known Aharonov-Bohm effect. In addi-tion, they discussed the Landau levels in the presence of acosmic string within the framework of Kaluza-Klein theory.In Ref. [31], the Klein-Gordon oscillator on the curved back-ground within the Kaluza-Klein theory was studied. Theproblem of the interaction between particles coupled har-monically with topological defects in the Kaluza-Klein theorywas studied. They considered a series of topological defectsand then treated the Klein-Gordon oscillator coupled to thisbackground, and obtained the energy eigenvalue and corre-sponding eigenfunctions in these cases. They have shownthat the energy eigenvalue depends on the global parameterscharacterizing these space-times. In Ref. [32], a scalar particlewith position-dependent mass subject to a uniform magneticfield and a quantum magnetic flux, both coming from thebackground which is governed by the Kaluza-Klein theory,were investigated. They inserted a Cornell-type scalar poten-tial into this relativistic systems and determined the relativis-tic energy eigenvalue of the system in this background ofextra dimension. They analyzed particular cases of this sys-tem, and a quantum effect was observed: the dependence ofthe magnetic field on the quantum numbers of the solutions.In Ref. [34], the relativistic quantum dynamics of a scalarparticle subject to linear potential on the curved backgroundwithin the Kaluza-Klein theory was studied. We have solvedthe generalized Klein-Gordon oscillator in the cosmic stringandmagnetic cosmic string space-time with a linear potentialwithin the Kaluza-Klein theory. We have shown that the

energy eigenvalues obtained there depend on the globalparameters characterizing these space-times and the gravita-tional analogue to the Aharonov-Bohm effect for boundstates [18–21, 26, 29] of a scalar particle was analyzed.

In this work, we have investigated the relativistic quan-tum dynamics of a scalar particle interacting with gravita-tional fields produced by topological defects via the Klein-Gordon oscillator of the Klein-Gordon equation in the pres-ence of cosmic string and magnetic cosmic string within theKaluza-Klein theory with scalar potential. We have deter-mined the manner in which the nontrivial topology due tothe topological defects and a quantummagnetic flux modifiesthe energy spectrum and wave-functions of a scalar particle.We then have studied the quantum dynamics of a scalar par-ticle interacting with fields by introducing a magnetic fluxthrough the line element of a cosmic string space-time usingthe five-dimensional version of the general relativity. Thequantum dynamics in the usual as well as magnetic cosmicstring cases allow us to obtain the energy eigenvalues andcorresponding wave-functions that depend on the externalparameters characterize the background space-time, a resultknown by a gravitational analogue of the well studiedAharonov-Bohm effect.

In section 2, we have chosen a Cornell-type functionf ðrÞ = a r + ðb/rÞ and Cornell-type potential SðrÞ = ηL r +ðηc/rÞ into the relativistic systems. We have solved the gen-eralized Klein-Gordon oscillator in the cosmic string back-ground within the Kaluza-Klein theory and obtained theenergy eigenvalues Eq. (23). We have plotted graphs ofthe energy eigenvalues Eq. (23) w. r. t. different parametersby Figures 1–5. By imposing the additional recurrence con-dition cn+1 = 0 on the relativistic eigenvalue problem, forexample n = 1, we have obtained the ground state energylevels and wave-functions by Eqs. (26)–(27). We have dis-cussed a special case corresponds to ηL ⟶ 0 and obtainedthe relativistic energy eigenvalues Eq. (36) of a generalizedKlein-Gordon oscillator in the cosmic string space-timewithin the Kaluza-Klein theory. We have also obtained therelativistic energy eigenvalues Eq. (43) of a free-scalar parti-cle by solving the Klein-Gordon equation with a Coulomb-type scalar potential in the background of cosmic stringspace-time in the Kaluza-Klein theory.

In section 3, we have studied the relativistic quantumdynamics of a scalar particle in the background of magneticcosmic string in the Kaluza-Klein theory with a scalar poten-tial. By choosing the same function f ðrÞ = a r + ðb/rÞ and aCoulomb-type scalar potential SðrÞ = ηc/r, we have solvedthe radial wave-equation in the considered system andobtained the bound states energy eigenvalues Eq. (58). Wehave plotted graphs of the energy eigenvalues Eq. (58) w. r.t. different parameters by Figures 6–9. Subsequently, theground state energy levels Eq. (59) and correspondingwave-functions Eq. (60) for the radial mode n = 1 by impos-ing the additional condition cn+1 = 0 on the eigenvalue prob-lem is obtained. Furthermore, a special case corresponds toa⟶ 0, b⟶ 0 is discussed and obtained the relativisticenergy eigenvalues Eq. (66) of a scalar particle by solvingthe Klein-Gordon equation with a Coulomb-type scalarpotential in the magnetic cosmic string space-time in the


Kaluza-Klein theory. For α⟶ 1, we have seen that theenergy eigenvalues Eq. (66) reduces to the result obtainedin Ref. [30]. As there is an effective angular momentumquantum number, l⟶ lef f = ð1/αÞ ðl − qΦ/2πÞ; thus, therelativistic energy eigenvalues Eqs. (58) and (66) depend onthe geometric quantum phase [21]. Hence, we have that En,lðΦ +Φ0Þ = En,l∓τðΦÞ where Φ0 = ±ð2π/qÞ τ with τ = 0, 1, 2, :.This dependence of the relativistic energy eigenvalues onthe geometric quantum phase gives rise to a relativistic ana-logue of the Aharonov-Bohm effect for bound states [19–21, 29]. Finally, we have obtained the persistent currents byEqs. (71)–(73) for this relativistic quantum system becauseof the dependence of the relativistic energy eigenvalues onthe geometric quantum phase.

So in this paper, we have shown some results which are inaddition to those results obtained in Refs. [18, 29–34] pre-sents many interesting effects.

Appendix

A. Brief Review of the Nikiforov-Uvarov(NU) Method

The Nikiforov-Uvarov method is helpful in order to findeigenvalues and eigenfunctions of the Schrödinger like equa-tion, as well as other second-order differential equations ofphysical interest. According to this method, the eigenfunc-tions of a second-order differential equation [94]

d2ψ sð Þds2

+ α1 − α2 sð Þs 1 − α3 sð Þ

dψ sð Þds

+ −ξ1 s2 + ξ2 s − ξ3� �

s2 1 − α3 sð Þ2 ψ sð Þ = 0:

ðA:1Þ

are given by

ψ sð Þ = sα12 1 − α3 sð Þ−α12− α13/α3ð Þ P α10−1, α11/α3ð Þ−α10−1ð Þn 1 − 2 α3 sð Þ:

ðA:2Þ

And that the energy eigenvalues equation

α2 n − 2 n + 1ð Þ α5 + 2 n + 1ð Þ ffiffiffiffiffiα9

p + α3ffiffiffiffiffiα8

pð Þ+ n n − 1ð Þ α3 + α7 + 2 α3 α8 + 2 ffiffiffiffiffiffiffiffiffiffi

α8 α9p = 0:

ðA:3Þ

The parameters α4,⋯, α13 are obtained from the sixparameters α1,⋯, α3 and ξ1,⋯, ξ3 as follows:

α4 =12 1 − α1ð Þ,

α5 =12 α2 − 2 α3ð Þ,

α6 = α25 + ξ1,

α7 = 2 α4 α5 − ξ2,

α8 = α24 + ξ3,

α9 = α6 + α3 α7 + α23 α8,

α10 = α1 + 2 α4 + 2 ffiffiffiffiffiα8

p ,

α11 = α2 − 2 α5 + 2 ffiffiffiffiffiα9


pð Þ,

α12 = α4 +ffiffiffiffiffiα8

p ,

α13 = α5 −ffiffiffiffiffiα9


pð Þ:ðA:4Þ

A special case where α3 = 0, as in our case, we find

limα3→0

P α10−1, α11/α3ð Þ−α10−1ð Þn 1 − 2 α3 sð Þ = Lα10−1n α11 sð Þ,

limα3→0

1 − α3 sð Þ−α12− α13/α3ð Þ = eα13 s:ðA:5Þ

Therefore, the wave-function from (2) becomes

ψ sð Þ = sα12 eα13 s Lα10−1n α11 sð Þ, ðA:6Þ

where LðαÞn ðxÞ denotes the generalized Laguerre polynomial.The energy eigenvalues equation reduces to

n α2 − 2 n + 1ð Þ α5 + 2 n + 1ð Þ ffiffiffiffiffiα9

p + α7 + 2 ffiffiffiffiffiffiffiffiffiffiα8 α9

p = 0:ðA:7Þ

Noted that the simple Laguerre polynomial is the specialcase α = 0 of the generalized Laguerre polynomial:

L 0ð Þn xð Þ = Ln xð Þ: ðA:8Þ

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interestregarding publication of this paper.

Acknowledgments

The author sincerely acknowledges the anonymous kind ref-eree(s) for their valuable comments and suggestions andthanks the editor.

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