# Analytical Solution of the PDM-Coulombic Klein-Gordon Oscillator in Kaluza-
Klein Theory

Abdelmalek Bouzenada<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>Laboratoire de Physique Appliquée et Théorique
Université Larbi-Tébessi-, Tébessa, Algeria  Abdelmalek Boumali
<EMAIL_ADDRESS>Laboratoire de Physique Appliquée et Théorique
Université Larbi-Tébessi-, Tébessa, Algeria  Omar Mustafa
<EMAIL_ADDRESS>Department of Physics, Eastern Mediterranean
University,
G. Magusa, north Cyprus, Mersin 10 - Turkey  R. L. L .Vitória
<EMAIL_ADDRESS><EMAIL_ADDRESS>Faculdade de Física,
Universidade Federal do Pará, Av. Augusto Corrêa,
Guamá, Belém, PA 66075-110, Brazil.  Marwan Al-Raeei<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>Faculty of Science,Damascus
University,Damascus,Syria

(August 28, 2024)

###### Abstract

In this contribution, the relativistic quantum motions of the position-
dependent mass (PDM) oscillator field with a scalar potential in the context
of the Kaluza-Klein theory is investigated. Through a purely analytical
analysis, the eigensolutions of this system have been obtained. The results
showed that the KGO is influenced not only by curvature, torsion and the
quantum flux associated with the extra dimension, but also by the possibility
of modifying the mass term of the Klein-Gordon equation.

Klein-Gordon oscillator, topological defects, Kalluza-Klein Theory, PDM ,
Coulomb-Type Potentiel .

###### pacs:

03.65.Ge; 03.50.−z; 05.70.Ce ; 03.65.−w ; 04.62.+v; 04.40.−b; 04.20.Gz;
04.20.Jb; 04.20.−q; 03.65.Pm

##  I Introduction

One of the fields of investigation in the context of quantum mechanics is the
gravitational effects on the relativistic quantum dynamics of a particle
characterized by well-defined spins, for example, spin-$1/2$ and spin-$0$
particles, described by the Dirac equation and the Klein-Gordon equation,
respectively, both extended to incorporate the curvature effects intrinsic to
the gravitation of the environment considered as background bd .

In particular, spin-0 particles have been heavily investigated subject to
gravitational effects in various types of space-time, for two reasons:
searches for interpretations or extensions of standard models of particle
physics, mainly due to the advent of the Higgs boson with its experimental
evidence and its powerful effect in the Standard Models ryder , as well as in
the description of particles or quasi-particles in condensed matter systems in
analogous models kleinert , such as the case of phonons bas , two-dimensional
systems dantas  and quantum dynamics in graphene sheets graff ; graff1  with
various defects generated by the process to cut and paste kat ; kat1 .

Spin-$0$ particles have been investigated, for example, in cosmic string
spacetime, subject to central potentials and external fields. We therefore
have studies of particles with position-dependent mass (PDM) subject to a
scalar and electromagnetic Coulomb-type potential, linear potential, Landau
quantization, Aharanov-Bohm effect (ABE) for bound states eug  and interacting
with the Klein-Gordon oscillator bou . It is noteworthy that the presence of
the cosmic string in the background is associated with the curvature of
spacetime vil ; sc .

Another ingredient considered in the background in order to extend the
understanding of bosonic spin-$0$ quantum dynamics is the presence of torsion
in the environment put . From a mathematical point of view, torsion in
spacetime arises due to the non-symmetry of the Christoffel symbols, thus
producing a tensor field associated with torsion kat ; kat1 ; put . However,
torsion can be typified according to the conservation of the geometric
symmetry produced after the "cut and paste" process kat ; put . The best-known
examples are time-type, spiral-type, and screw-type dislocations kat1 ; put .
In particular, from the point of view of condensed matter, the screw-like
dislocation is associated with the Burgers vector kat ; put . Spin-$0$
particle has been investigated in spacetime with torsion, for example,
subjected to ABE for bound states ric , by interacting to electromagnetic and
scalar Coulomb-type potential ric1 , linear potential and Klein-Gordon
oscillator ric2 . In addition, there are studies of scalar particles immersed
in the spacetime with a screw-type dislocation under effects of rotation ric3
; ric4 .

A quantum effect that can be observed of cosmic string and screw-like
dislocation on bound state solutions in quantum particle dynamics: effect
analogous to ABE for bound states abe ; abe1 . That is, the quantum numbers
associated with the angular momentum are modified by the background topology,
generating a kind of "correction" in the eigenvalues of the angular momentum,
which, in turn, provides an effective angular momentum valdir ; valdir1 ;
boumali3 ; boumali4 ; boumali5 .

In particular, ABE for bound states has been investigated in a space-time with
an extra dimension erico , known as the Kaluza-Klein theory (KKT) kkt ; kkt1 .
and quantum mechanics is possible, but with the insertion of extra dimensions.
This proposal is defended in the famous string theory kkt2 . Although KKT is
much simpler than more modern theories of extra dimensions, it has been widely
used for research into the relativistic quantum dynamics of fundamental
particles kkt2 ; bou . In the case of spin-$0$ particles, we have studies in
systems with PDM erico1 , Landau quantization erico2 , Klein-Gordon oscillator
erico3  and under effects of rotation erico4 .

Therefore, in this analysis, we intend to investigate the effects of curvature
vil  and torsion put  of space-time incremented with an extra dimension on an
electrically charged scalar quantum oscillator subjected to a quantum flux
analogous to the ABE. Our analysis will be purely analytical in obtaining the
relativistic energy profiles, as well as in determining the axial
eigenfunctions that describe the oscillations of the electrically charged
scalar particle in this non-trivial spacetime.

The structure of this paper is as follows: in Sec. 2, we make a brief review
on quantum dynamics of a free particle in KKT spacetime with curvature and
torsion, in which we show that the general solution of the scalar particle is
defined through the Bessel function of the first type redefined in terms of
the parameters associated with the curvature and torsion of KKT space-time; in
Sec. 3, we investigated the effects of central linear potential in the mass of
the quantum oscillator immersed in the KKT space-time, where we determined
bound state solutions; in Sec. 4 we continue with the insertion of central
potentials in the mass term of the modified Klein-Gordon equation, in which
the potential considered is a Coulomb-type potential, in which we define the
relativistic energy profile of the quantum oscillator; in Sec. 5, we present
our conclusions.

##  II An overview of The Klein-Gordon oscillators in a cosmic string within
Kaluza-Klein Theory

The purpose of this section is to investigate the dynamics of the Klein-Gordon
oscillator (KGO) in the framework of Kaluza-Klein theory geometry. It is
widely known that the relativistic wave equations for a scalar particle in a
Riemannian spacetime, described by the metric tensor $g_{\mu\nu}$, can be
obtained through a reformulation of the Klein-Gordon (KG) equation. This
provides an opportunity to examine the behavior of the KGO in the presence of
cosmic strings (key-B1, ; key-B2, ).

$\left(\square+m^{2}-\mathcal{\xi}R\right)\Psi(x,t)=0,$ (1)

In this context, the symbol $\square$ denotes the Laplace-Beltrami operator,
which is commonly defined as follows:

$\square=g^{\mu\nu}D_{\mu}D_{\nu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right),$
(2)

In the provided equation, the parameter $\xi$ corresponds to a dimensionless
coupling constant with real values. The Ricci scalar curvature, denoted as
$R$, can be expressed as $R=g^{\mu\nu}R_{\mu\nu}$, where $R_{\mu\nu}$ denotes
the Ricci curvature tensor. The inverse of the metric tensor is represented by
$g^{\mu\nu}$, and the determinant of the metric tensor is denoted as $g$,
specifically $g=\det\left(g_{\mu\nu}\right)$.

Our subsequent aim is to investigate the quantum dynamics of spin-0 particles
within the space-time generated by a $(4+1)$-dimensional space. In this
context, we will explore the behavior and properties of these particles in
this extended dimensional framework.

###  II.1 Free Klein-Gordon equation in the background of a cosmic string in a
Kaluza–Klein theory

In this section (key-B3, ), we embark on an exploration of a fundamental
topological defect that serves as the cornerstone of our research. Taking
inspiration from the intriguing characteristics observed in edge dislocations
present within crystalline solids, we extend the concept of such defects into
the realm of gravity. Upon examining an edge dislocation, we discern its
remarkable metamorphosis from a circular form into a spiral dislocation. The
line element that elucidates the space-time curvature incorporating this
significant topological defect is expressed in units where $\hbar=c=1$
(key-B3, ; key-B4, ). By establishing this connection, we bridge the
principles governing crystal defects with the intricate structure of space-
time.

$\displaystyle ds^{2}$ $\displaystyle=g_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=dt^{2}-d\rho^{2}-\left(\alpha\rho\right)^{2}d\varphi^{2}-\left(dz+Jd\varphi\right)^{2}-\left(dx+\frac{\Phi}{2\pi}d\varphi\right)^{2},$
(3)

In this context, where $\chi$ represents a constant value associated with the
distortion of the defect, the parameter
$J=\frac{\left|\overrightarrow{b}\right|}{2\pi}$ also relates to the Burgers
vector $\overrightarrow{b}$. The coordinate ranges are as follows:
$-\infty\leq t\leq+\infty$, $r\geq 0$, $0\leq\varphi\leq 2\pi$, $-\infty\leq
z,x\leq+\infty$, and $\alpha\in[0,1[$. The angular parameter $\alpha$
determines the angular deficit $\delta\varphi=2\pi(1-\alpha)$, which is
connected to the linear mass density $\mu$ of the string through
$\alpha=1-4\mu$. It is important to note that this metric provides an accurate
solution to Einstein’s field equations for $0\leq\mu<1/4$. Furthermore, by
setting $\varphi^{\prime}=\alpha\varphi$, it represents a flat conical outer
space with an angular deficit $\delta\phi=8\pi\mu$.

When we consider the components of the metric tensor and its inverse, denoted
as $\left(g_{\mu\nu}\right)$ and $\left(g^{\mu\nu}\right)$, respectively
(key-B3, ; key-B4, ), we find the following expressions:

$g_{\mu\nu}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&-1&0&0&0\\\
0&0&-\left(\left(\alpha\rho\right)^{2}+\left(\frac{\Phi}{2\pi}\right)^{2}+J^{2}\right)&-J&-\frac{\Phi}{2\pi}\\\
0&0&-J&-1&0\\\ 0&0&-\frac{\Phi}{2\pi}&0&-1\end{array}\right),$ (4)

and

$g^{\mu\nu}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&-1&0&0&0\\\
0&0&-\frac{1}{\left(\alpha\rho\right)^{2}}&\frac{J}{\left(\alpha\rho\right)^{2}}&\frac{\Phi}{2\pi\left(\alpha\rho\right)^{2}}\\\
0&0&\frac{J}{\left(\alpha\rho\right)^{2}}&-\left(1+\frac{J^{2}}{\left(\alpha\rho\right)^{2}}\right)&-\frac{\Phi
J}{2\pi\left(\alpha\rho\right)^{2}}\\\
0&0&\frac{\Phi}{2\pi\left(\alpha\rho\right)^{2}}&-\frac{\Phi
J}{2\pi\left(\alpha\rho\right)^{2}}&-\left(1+\frac{\Phi^{2}}{\left(2\pi\alpha\rho\right)^{2}}\right)\end{array}\right)$
(5)

where

$\psi(t,\rho,\varphi,z,x)=\psi(\rho)e^{-i\left(Et-l\varphi-Kz-\lambda
x\right)},$ (6)
$\left[\frac{d^{2}}{d\rho^{2}}+\frac{1}{\rho}\frac{d}{d\rho}-\frac{\zeta^{2}}{\rho^{2}}+\kappa^{2}\right]\psi\left(\rho\right)=0,$
(7)

where we have set

$\displaystyle\zeta=$
$\displaystyle\frac{\sqrt{\left(l-JK\right)^{2}+\frac{\Phi^{2}}{\pi^{2}}\left[\frac{\lambda^{2}}{4}-\frac{l\lambda}{\Phi}+\frac{JK\lambda\pi}{\Phi}\right]}}{\alpha}$
(8) $\displaystyle\kappa=$ $\displaystyle\sqrt{E^{2}-m^{2}-K^{2}-\lambda^{2}}$

The equation represented by Eq. (7) can be identified as a Bessel equation.
Its general solution is defined as follows:

$\psi\left(\rho\right)=A\,J_{|\zeta|}\left(\kappa\rho\right)+B\,Y_{|\zeta|}\left(\kappa\rho\right),$
(9)

where $J_{|\zeta|}\left(\kappa\rho\right)$ and
$Y_{|\zeta|}\left(\kappa\rho\right)$ are the Bessel functions of order $\zeta$
and of the first and the second kind, respectively. Here $A$ and $B$ are
arbitrary constants. We notice that at the origin when $\zeta=0$, the function
$J_{|\zeta|}\left(\kappa\rho\right)\neq 0$. However,
$Y_{|\zeta|}\left(\kappa\rho\right)$ is always divergent at the origin. In
this case, we will consider only $J_{|\zeta|}\left(\kappa\rho\right)$ when
$\zeta\neq 0$. Hence, we write the solution to Eq. (7) as follows

$\psi\left(\rho\right)=A\,J_{\frac{\left|\sqrt{\left(l-JK\right)^{2}+\frac{\Phi^{2}}{\pi^{2}}\left[\frac{\lambda^{2}}{4}-\frac{l\lambda}{\Phi}+\frac{JK\lambda\pi}{\Phi}\right]}\right|}{\alpha}}\,\,\left(\sqrt{E^{2}-m^{2}-K^{2}-\lambda^{2}}\,\rho\right),$
(10)

We can now express the wavefunction of the spinless heavy KG particle in the
space-time of a cosmic dislocation using this solution.

$\psi\left(t,\rho,\varphi,z,x\right)=\left|\mathcal{N}_{1}\right|e^{-i\left(Et-l\varphi-
Kz-\lambda
x\right)}\,J_{\frac{\left|\sqrt{\left(l-JK\right)^{2}+\frac{\Phi^{2}}{\pi^{2}}\left[\frac{\lambda^{2}}{4}-\frac{l\lambda}{\Phi}+\frac{JK\lambda\pi}{\Phi}\right]}\right|}{\alpha}}\,\,\left(\sqrt{E^{2}-m^{2}-K^{2}-\lambda^{2}}\,\rho\right),$
(11)

The determination of the constant $\left|\mathcal{N}_{1}\right|$ can be
achieved by applying the necessary normalization condition to the Klein-Gordon
equation. However, it is fortunate that the inability to determine the
normalization constants in this manuscript does not impact the final outcomes
or results.

###  II.2 Klein-Gordon oscillator in the background of a cosmic string in a
Kaluza–Klein theory

In order to proceed, it is necessary to substitute the momentum operator into
Equation (1). Consequently, Equation (10) can be reformulated as follows.
Similarly, employing a straightforward calculation based on the aforementioned
approach, we can derive the following differential equation (key-B3, ).

$\left[\frac{d^{2}}{d\rho^{2}}+\frac{1}{\rho}\frac{d}{d\rho}-m^{2}\omega^{2}\rho^{2}-\frac{\sigma^{2}}{\rho^{2}}+E^{2}+\delta\right]\psi\left(\rho\right)=0,$
(12)

with

$\displaystyle\sigma^{2}=$
$\displaystyle\left(\frac{\left(l-JK\right)^{2}+\frac{\Phi^{2}}{\pi^{2}}\left[\frac{\lambda^{2}}{4}-\frac{l\lambda\pi}{\Phi}+\frac{JK\lambda\pi}{\Phi}\right]}{\alpha}\right)^{2}$
(13)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta-E^{2}}=$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}2m\omega-m^{2}-K^{2}-\lambda^{2}$

The Klein-Gordon oscillator (KGO) equation for a spin-0 particle within the
(1+4) space-time of Kaluza-Klein theory is represented by Equation (12). In
order to obtain the solution to this problem, we propose a radial coordinate
transformation as a preliminary step.

$\mathcal{S}=m\omega r^{2},$ (14)

subsitutuing the expression for $\chi$ into Eq. (12), we obtain

$\left[\frac{d^{2}}{d\mathcal{\mathcal{S}}^{2}}+\frac{1}{\mathcal{S}}\frac{\partial}{d\mathcal{\mathcal{S}}}-\frac{\sigma^{2}}{4\mathcal{\mathcal{S}}^{2}}+\frac{\delta}{4m\omega\mathcal{\mathcal{S}}}-\frac{1}{4}\right]\psi\left(\mathcal{S}\right)=0.$
(15)

By examining the asymptotic behavior of the wave function at both the origin
and infinity, and aiming to find regular solutions, we can consider a solution
of the following form:

$\psi\left(\mathcal{\mathcal{S}}\right)=\mathcal{\mathcal{S}}^{\frac{\left|\sigma\right|}{2}}e^{-\frac{\mathcal{\mathcal{S}}}{2}}F\left(\mathcal{\mathcal{S}}\right),$
(16)

As previously, we can plug this back into Eq. (15), and we get

$\mathcal{S}\frac{d^{2}F\left(\mathcal{S}\right)}{d\mathcal{S}^{2}}+\left(|\sigma|+1-\chi\right)\frac{dF\left(\mathcal{S}\right)}{d\mathcal{S}}-\left(\frac{|\sigma|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}\right)F\left(\mathcal{S}\right)=0,$
(17)

The equation at hand corresponds to the confluent hypergeometric equation
(key-64), and its solutions are expressed in terms of a specific type of
confluent hypergeometric function.

$F\left(\mathcal{\mathcal{S}}\right)=_{1}F_{1}\left(\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},+1,\mathcal{\mathcal{S}}\right),$
(18)

It is important to mention that the solution given by Equation (18) must be a
polynomial function of degree $n$. However, as we let $n\rightarrow\infty$, a
divergence issue arises. In order to have a finite polynomial, the factor
associated with the last term in Equation (17) must be a negative integer.
This condition implies that:

$\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}=-n_{r}\qquad,n_{r}=0,1,2.....,$
(19)

Utilizing this result along with the parameters given in Equation (14), we can
derive the quantized energy spectrum of the Klein-Gordon oscillator (KGO) in
the space-time of a cosmic dislocation. Consequently, we are able to determine
the following:

$E^{\pm}\left(n_{r}\right)=\pm\sqrt{4m\omega
n_{r}+\frac{2m\omega}{\alpha}\left|\left(l-JK\right)^{2}+\frac{\Phi^{2}}{\pi^{2}}\left[\frac{\lambda^{2}}{4}-\frac{l\pi\lambda}{\Phi}+\frac{JK\lambda\pi}{\Phi}\right]\right|+m^{2}+K^{2}+\lambda^{2}},$
(20)

It is evident that the energy is directly dependent on the angular deficit
$\alpha$. In other words, the presence of the wedge angle, which is caused by
the topological defect (i.e., the cosmic string), influences the curvature of
the space-time. As a result, it affects the relativistic dynamics of the
scalar particle by introducing a gravitational field. The corresponding wave
function is defined as follows:

$\psi\left(\rho\right)=\left|\mathcal{N}_{2}\right|\left(m\omega\rho^{2}\right)^{\frac{\left|\sigma\right|}{2}}e^{-\frac{m\omega\rho^{2}}{2}}{}_{1}F_{1}\left(\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},\sigma+1,m\omega\rho^{2}\right),$
(21)

Subsequently, the general eigenfunctions are expressed as:

$\psi\left(t,\rho,\varphi,z,x\right)=\left|\mathcal{N}_{2}\right|\left(m\omega\rho^{2}\right)^{\frac{\left|\sigma\right|}{2}}e^{-\frac{m\omega\sigma^{2}}{2}}e^{-i\left(Et-l\varphi-
Kz-\lambda
x\right)}{}_{1}F_{1}\left(\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},\sigma+1,m\omega\rho^{2}\right),$
(22)

Here, $\left|\mathcal{N}_{2}\right|$ represents the normalization constant,
and the values of $\left(\sigma,\delta\right)$ are provided in Equation (13).

##  III PDM of Klein-Gordon oscillator in the background of a cosmic string in
a Kaluza-Klein theory

In this subsection, we shall use the non-minimal coupling form of the PDM-
momentum operator

$\mathbf{\hat{p}}\left(\mathbf{r}\right)=-i\left(\mathbf{\nabla-}\frac{\mathbf{\nabla}f\left(\mathbf{r}\right)}{4f\left(\mathbf{r}\right)}\right)\Longleftrightarrow
p_{j}=-i\left(\partial_{j}-\frac{\partial_{j}f\left(\mathbf{r}\right)}{4f\left(\mathbf{r}\right)}\right),$
(23)

introduced by Mustafa and Algadhi O1 . It has been shown that this PDM-
momentum operator effectively yields the what is known in the literature as
Mustafa and Mazharimousavi’s ordering O2 ; O3  of the von Roos PDM-kinetic
energy Schrödinger operator O4 . It has been used in the study of PDM Klein-
Gordon (KG) particles in different spacetime backgrounds O5 ; O6 ; O7 ; O8 ;
O9 . Under such non-minimal coupling form, our PDM operator reads

$\tilde{p}_{\mu}\longrightarrow-i\partial_{\mu}+i\mathcal{F}_{\mu}\,;\,\,\mathcal{F}_{\mu}=\left(0,\mathcal{F}_{\rho},0,0\right),\,\,\mathcal{F}_{\rho}=\frac{f^{\prime}\left(\rho\right)}{4f\left(\rho\right)}$
(24)

and consequently our PDM KG-equation reads

$\frac{1}{\sqrt{-g}}\left(D_{\mu}+\mathcal{F}_{\mu}\right)\left[\sqrt{-g}g^{\mu\nu}\left(D_{\nu}-\mathcal{F}_{\nu}\right)\Psi\right]=\left(m+S\left(\rho\right)\right)^{2}\Psi.$
(25)

where $D_{\mu}=\partial_{\mu}-ieA_{\mu}$ is the covariant derivative and
$S\left(\rho\right)$ is the Lorentz scalar radial potential. This would, with
$\Psi$ given in (11), imply that

$\psi^{\prime\prime}\left(\rho\right)+\frac{1}{\rho}\psi^{\prime}\left(\rho\right)+\left[\mathcal{E}^{2}-\frac{\tilde{\zeta}^{2}}{\rho^{2}}-\mathcal{M}\left(\rho\right)-2mS\left(\rho\right)-S\left(\rho\right)^{2}\right]\psi\left(\rho\right)=0,$
(26)

where $\mathcal{E}^{2}=E^{2}-\left(K^{2}+\lambda^{2}+m^{2}\right)$, and

$\tilde{\zeta}^{2}=\frac{1}{\alpha^{2}}\left\\{\left(\left[\ell-
eA_{\varphi}\right]-JK\right)^{2}+\lambda\,\bar{\phi}\left[\lambda\,\bar{\phi}-2\left(\ell-
eA_{\varphi}\right)+2JK\right]\right\\};\bar{\phi}=\frac{\phi}{2\pi},$ (27)

and

$\mathcal{M}\left(\rho\right)=\mathcal{F}_{\rho}^{\prime}+\frac{\mathcal{F}_{\rho}}{\rho}+\mathcal{F}_{\rho}^{2}.$
(28)

The substitution of $\psi\left(\rho\right)=U\left(\rho\right)/\sqrt{\rho}$
would imply

$U^{\prime\prime}\left(\rho\right)+\left[\mathcal{E}^{2}-\frac{\left(\tilde{\zeta}^{2}-1/4\right)}{\rho^{2}}-\mathcal{M}\left(\rho\right)-2mS\left(\rho\right)-S\left(\rho\right)^{2}\right]U\left(\rho\right)=0.$
(29)

Eventually, with $\mathcal{F}_{\rho}=\eta\rho$ (or if you wish,
$f\left(\rho\right)=\exp\left(2\eta\rho\right)$) of (24), we obtain

$U^{\prime\prime}\left(\rho\right)+\left[\mathcal{\tilde{E}}^{2}-\frac{\left(\tilde{\zeta}^{2}-1/4\right)}{\rho^{2}}-\mathcal{\eta}^{2}\rho^{2}-2mS\left(\rho\right)-S\left(\rho\right)^{2}\right]U\left(\rho\right)=0.$
(30)

where $\mathcal{\tilde{E}}^{2}=\mathcal{E}^{2}-2\eta$. At this point, one may
observe that for $\eta=m\omega$, $A_{\varphi}=0$, and $S\left(\rho\right)=0$,
the case discussed for (12) is retrieved to obtain

$\mathcal{\tilde{E}}^{2}=2\eta\left(2n_{r}+\left|\tilde{\zeta}\right|+1\right)\Leftrightarrow
E=\pm\sqrt{2\eta\left(2n_{r}+\left|\tilde{\zeta}\right|+2\right)+K^{2}+\lambda^{2}+m^{2}},$
(31)

and

$U\left(\rho\right)\sim\rho^{\left|\tilde{\zeta}\right|+1/2}\exp\left(-\frac{\eta\rho^{2}}{2}\right)L_{n_{r}}^{\left|\tilde{\zeta}\right|}\left(\left|\tilde{\zeta}\right|\rho^{2}\right).$
(32)

###  III.1 KG-oscillators plus linear confinement $S\left(\rho\right)=A\rho$

With a linear Lorentz scalar potential $S\left(\rho\right)=A\rho$, equation
(30) reads

$U^{\prime\prime}\left(\rho\right)+\left[\mathcal{\tilde{E}}^{2}-\frac{\left(\tilde{\zeta}^{2}-1/4\right)}{\rho^{2}}-\mathcal{\tilde{\eta}}^{2}\rho^{2}-2mA\rho\right]U\left(\rho\right)=0\;;\;\mathcal{\tilde{\eta}}^{2}=\mathcal{\eta}^{2}+A^{2}.$
(33)

Hereby, we shall be interested in an exact solution for equation (33) for
$\tilde{\zeta}=1/2\Leftrightarrow 4\left(\ell-
JK-\lambda\bar{\phi}\right)^{2}=\alpha^{2}$, so that equation (33) reads

$\left\\{\partial_{\rho}^{2}-\mathcal{\tilde{\eta}}^{2}\rho^{2}-2mA\,\rho+\mathcal{\tilde{E}}^{2}\right\\}U\left(\rho\right)=0.$
(34)

At this point, one should be aware that this equation resembles the radially
spherically symmetric Schrödinger equation with an irrational angular momentum
quantum number $\tilde{\ell}=\tilde{\zeta}-1/2=0$ in the central
attractive/repulsive core $\tilde{\ell}(\tilde{\ell}+1)/\rho^{2}$. Let us use
a radial wave function in the form of

$U\left(\rho\right)=\exp\left(-\frac{|\tilde{\eta}|\rho^{2}}{2}-\frac{mA\rho}{|\tilde{\eta}|}\right)\,\left[B_{1}\,(\rho+\frac{mA}{\mathcal{\tilde{\eta}}^{2}})\,F(\rho)+B_{2}\,G(\rho)\right];\,\,\tilde{A}=\frac{mA}{\mathcal{\tilde{\eta}}^{2}}.$
(35)

Obviously, the exponential term in (35) represents the asymptotic behaviour
$U(\rho)\rightarrow 0$ as $\rho\rightarrow\infty$. Whereas, the asymptotic
behaviour of $U(\rho)$ as $\rho\rightarrow 0$ is either finite or zero,
depending on the effective interaction potential at hand ( e.g., O10 ). We
shall instead let the general functions $F(\rho)$ and $G(\rho)$ have their say
in the process. Hence, the substitution of $U(\rho)$, (35), would in a
straightforward manner result in

$\displaystyle
B_{1}\,\left[(\rho+\tilde{A})\,\partial_{\rho}^{2}-2|\tilde{\eta}|\left((\rho+\tilde{A})^{2}-\frac{1}{|\tilde{\eta}|}\right)\,\partial_{\rho}+[\mathcal{\tilde{E}}^{2}-3|\tilde{\eta}|+\frac{m^{2}A^{2}}{\tilde{\eta}^{2}}]\right]F(\rho)$
$\displaystyle+\,B_{2}\,\left[(\rho+\tilde{A})\,\partial_{\rho}^{2}-2|\tilde{\eta}|((\rho+\tilde{A})\,\partial_{\rho}+[\mathcal{\tilde{E}}^{2}-|\tilde{\eta}|+\frac{m^{2}A^{2}}{\tilde{\eta}^{2}}]\right]G(\rho)=0.$
(36)

This equation suggests that

$\left[(\rho+\tilde{A})\,\partial_{\rho}^{2}-2|\tilde{\eta}|\left((\rho+\tilde{A})^{2}-\frac{1}{|\tilde{\eta}|}\right)\,\partial_{\rho}+[\mathcal{\tilde{E}}^{2}-3|\tilde{\eta}|+\frac{m^{2}A^{2}}{\tilde{\eta}^{2}}]\right]F(\rho)=0,$
(37)

and

$\left[(\rho+\tilde{A})\,\partial_{\rho}^{2}-2|\tilde{\eta}|((\rho+\tilde{A})\,\partial_{\rho}+[\mathcal{\tilde{E}}^{2}-|\tilde{\eta}|+\frac{m^{2}A^{2}}{\tilde{\eta}^{2}}]\right]G(\rho)=0,$
(38)

A change of variable a change of variable
$y=|\mathcal{\tilde{\eta}}|(\rho+\tilde{A})^{2}$ would imply

$\left[y\,\partial_{y}^{2}+(\frac{3}{2}-y)\,\partial_{y}+\frac{1}{4|\mathcal{\tilde{\eta}}|}[\mathcal{\tilde{E}}^{2}-3|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}]\right]\,F(y)=0,$
(39)

and

$\left[y\,\partial_{y}^{2}+(\frac{1}{2}-y)\,\partial_{y}+\frac{1}{4|\mathcal{\tilde{\eta}}|}[\mathcal{\tilde{E}}^{2}-|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}]\right]\,G(y)=0.$
(40)

One should be able to observe that both equations are in the form of Kummer’s
equation

$[u\,\partial_{u}^{2}+(b-u)\,\partial_{u}-a]W(u)=0.$ (41)

to immediately imply that

$F(y)=\,_{1}F_{1}(-\frac{1}{4|\mathcal{\tilde{\eta}}|}(\mathcal{\tilde{E}}^{2}-3|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}),\frac{3}{2},y),$
(42)

and

$G(\eta)=\,_{1}F_{1}(-\frac{1}{4|\mathcal{\tilde{\eta}}|}(\mathcal{\tilde{E}}^{2}-|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}),\frac{1}{2},y),$
(43)

Our general solution for (35) therefore reads

$\displaystyle U\left(\rho\right)$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{|\tilde{\eta}|\rho^{2}}{2}-\frac{mA\rho}{|\tilde{\eta}|}\right)\,\left[B_{1}\,(\rho+\frac{mA}{\mathcal{\tilde{\eta}}^{2}})\,_{1}F_{1}(-\frac{1}{4|\mathcal{\tilde{\eta}}|}(\mathcal{\tilde{E}}^{2}-3|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}),\frac{3}{2},y)\right.$
(44) $\displaystyle\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \
}\left.+B_{2}\,\,{}_{1}F_{1}(-\frac{1}{4|\mathcal{\tilde{\eta}}|}(\mathcal{\tilde{E}}^{2}-|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}),\frac{1}{2},y)\right]$

Now we have to appeal to textbook exact solvability of a pure harmonic
oscillator $\mathcal{\tilde{\eta}}^{2}\rho^{2}$, with $A=0$, and enforce the
condition that $U\left(\rho\right)=0$ for $\rho\rightarrow 0$. Consequently,
$B_{2}=0$ and our sought-after solution is obviously given, with
$y=|\mathcal{\tilde{\eta}}|(\rho+\tilde{A})^{2}$, by

$U\left(\rho\right)=B_{1}\,(\rho+\frac{mA}{\mathcal{\tilde{\eta}}^{2}})\,\exp\left(-\frac{|\tilde{\eta}|\rho^{2}}{2}-\frac{mA\rho}{|\tilde{\eta}|}\right)\,\,_{1}F_{1}(-\frac{1}{4|\mathcal{\tilde{\eta}}|}(\mathcal{\tilde{E}}^{2}-3|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}),\frac{3}{2},y).$
(45)

However, finiteness and square integrability of the quantum mechanics wave
functions requires that the confluent hypergeometric series should be
truncated into a polynomial of order $n_{r}=0,1,2,\cdots$. Under such
condition

$-n_{r}=-\frac{1}{4|\mathcal{\tilde{\eta}}|}(\mathcal{\tilde{E}}^{2}-3|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}})\Longrightarrow\mathcal{\tilde{E}}^{2}=\mathcal{E}^{2}-2\left|\eta\right|=2|\mathcal{\tilde{\eta}}|\left(2n_{r}+\frac{3}{2}\right)-\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}},$
(46)

and

$E=\pm\sqrt{2|\mathcal{\tilde{\eta}}|\left(2n_{r}+\frac{3}{2}\right)+2\left|\eta\right|+\left(K^{2}+\lambda^{2}+m^{2}\right)-\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}}.$
(47)

It should be noted that this results is in exact accord with that in (31) for
$\tilde{\zeta}=1/2$ and $A=0$. Our solution in (45) suggests, beyond doubt,
that a quantum particle moving in an oscillator-plus-linear radial potential
is allowed to reach the center at $\rho=0$ with some amplitude ( e.g., O10 ).
Then the conjecture that $U(\rho)\rightarrow 0$ as $\rho\rightarrow 0$ is not
a general admissible condition in quantum mechanics, but finiteness and square
integrability of the radial wave function is the only valid condition in
general.

On the other hand, it could be interesting to know that equation (33) admits a
solution in the form of biconfluent Heun functions

$U\left(\rho\right)=C_{1}\,\rho^{\left|\tilde{\zeta}\right|+1/2}\exp\left(-\frac{|\tilde{\eta}|\rho^{2}}{2}-\frac{mA\rho}{|\tilde{\eta}|}\right)H_{B}\left(\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime},\sqrt{|\tilde{\eta}|}\rho\right),$
(48)

where

$\alpha^{\prime}=2\left|\tilde{\zeta}\right|,\;\beta^{\prime}=\frac{2mA}{|\tilde{\eta}|^{3/2}},\;\gamma^{\prime}=\frac{m^{2}A^{2}+\mathcal{\tilde{E}}^{2}\tilde{\eta}^{2}}{\tilde{\eta}|^{3}},\;\delta^{\prime}=0.$
(49)

This would result, for $\tilde{\zeta}=1/2$,

$U\left(\rho\right)=C_{1}\,\rho\,\exp\left(-\frac{|\tilde{\eta}|\rho^{2}}{2}-\frac{mA\rho}{|\tilde{\eta}|}\right)H_{B}\left(1,\beta^{\prime},\gamma^{\prime},0,\sqrt{|\tilde{\eta}|}\rho\right).$
(50)

A simple comparison between (45) and (50) would suggest, up to a constant,
that

$(\rho+\frac{mA}{\mathcal{\tilde{\eta}}^{2}})\,\,\,_{1}F_{1}(-\frac{1}{4|\mathcal{\tilde{\eta}}|}(\mathcal{\tilde{E}}^{2}-3|\mathcal{\tilde{\eta}}|+\frac{m^{2}A^{2}}{\mathcal{\tilde{\eta}}^{2}}),\frac{3}{2},|\mathcal{\tilde{\eta}}|(\rho+\tilde{A})^{2})=\rho\,\,H_{B}\left(1,\beta^{\prime},\gamma^{\prime},0,\sqrt{|\tilde{\eta}|}\rho\right)$
(51)

to imply

$(x+\frac{\beta^{\prime}}{2})\,_{1}F_{1}\left(\frac{3-\gamma^{\prime}}{4},\,\frac{3}{2},(x+\frac{\beta^{\prime}}{2})^{2}\right)=x\,H_{B}(1,\beta^{\prime},\gamma^{\prime},0,x);\,\,x=\sqrt{|\tilde{\eta}|}\rho.$
(52)

Hence, the truncation of the bi-confluent Heun series
$H_{B}\left(\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime},\sqrt{|\tilde{\eta}|}\rho\right)$
into a polynomial of of degree $n\geq 0$, , one has to apply the condition
that
$\gamma^{\prime}=2\left(n+1\right)+\alpha^{\prime}\Longrightarrow\gamma^{\prime}=2\left(n+3/2\right)$
for $\alpha^{\prime}=1$. Whereas the truncation of the confluent geometric
series into a polynomial of order $n_{r}=0,1,2,\cdots$ suggests that
$-4n_{r}=3-\gamma^{\prime}\Longrightarrow\gamma^{\prime}=2(2n_{r}+3/2$.
Consequently, the confluent geometric and bi-confluent Heun polynomials are
both of even powers and $n=2n_{r}\geq 0$. Moreover, our result in (52) would,
for $\beta^{\prime}=0$, yield the correlation

$\,{}_{1}F_{1}\left(\frac{3-\gamma^{\prime}}{4},\,\frac{3}{2},x^{2}\right)=\,H_{B}(1,0,\gamma^{\prime},0,x),$
(53)

which is, in fact, in exact accord with that reported by Ronveaux O11

$H_{B}(\alpha^{\prime},0,\gamma^{\prime},0,y)=\,_{1}F_{1}\left(\frac{1}{2}+\frac{\alpha^{\prime}}{4}-\frac{\gamma^{\prime}}{4},1+\frac{\alpha^{\prime}}{2},y^{2}\right).$
(54)

##  IV  Free KG equation in the background of a cosmic string in a
Kaluza–Klein theory under coulomb-type potential

In this section, we are studying the interaction between the K-G equation and
Coulomb-type potential in KKT, in the space-time caused by a (4+1)-dimensional
stationary cosmic string described in Coulomb-Type Potentials
$\left(S\left(r\right)=\frac{\kappa}{r}=\pm\frac{\left|\kappa\right|}{r}\right)$(key-012,
).

$\left(\frac{d^{2}}{d\rho^{2}}+\frac{1}{\rho}\frac{d}{d\rho}-\frac{\sigma^{2}}{\rho^{2}}+\left(E-S\left(\rho\right)\right)^{2}-\xi\right)\psi\left(\rho\right)=0$
(55)

where

$\left(\frac{d^{2}}{d\rho^{2}}+\frac{1}{\rho}\frac{d}{d\rho}-\frac{\sigma^{2}}{\rho^{2}}+\left(E-\frac{\kappa}{\rho}\right)^{2}-\xi\right)\psi\left(\rho\right)=0$
(56)

and

$\xi=\left(m^{2}+K^{2}+\lambda^{2}\right)-\left(\frac{\Omega}{\alpha}\right)^{2}\left(\ell-
JK-\frac{\Phi}{2\pi}\lambda\right)^{2}$ (57)

By simplifying the last differential equation we find

$\left(\frac{d^{2}}{d\rho^{2}}+\frac{1}{\rho}\frac{d}{d\rho}-\frac{\left(\sigma^{2}-\kappa^{2}\right)}{\rho^{2}}-\frac{2\kappa
E}{\rho}+E^{2}-\xi\right)\psi\left(\rho\right)=0$ (58)

The solution to this differential equation is the following:

$\psi\left(\rho\right)=\mathcal{N}\rho^{\alpha}\mathrm{WhittakerM}\left[\beta,\gamma,\vartheta\,\rho\right]$
(59)

With derivation and compensation, we find the following constants

$\begin{array}[]{c}\alpha=-1/2\\\ \beta=-\frac{\kappa E}{\sqrt{\xi-E^{2}}}\\\
\gamma=\mathrm{i}\left(\sqrt{\kappa^{2}-\sigma^{2}}\right)\\\
\vartheta=2\sqrt{\xi-E^{2}}\end{array}$ (60)

The general solution of the last differential equation is given in the
following form:

$\psi\left(\rho\right)=\mathcal{N}\rho^{-\frac{1}{2}}\mathrm{WhittakerM}\left[-\frac{\kappa
E}{\sqrt{\xi-E^{2}}},\mathrm{i}\left(\sqrt{\kappa^{2}-\sigma^{2}}\right),2\sqrt{\xi-E^{2}}\,\rho\right]$
(61)

The mathematical relationship between the WhitakarM function and the
hypergeometric function ($1F1$) is given from the general form(key-012, ;
key-013, )

$\mathrm{WhittakerM}\left(\mu,\nu,z\right)=\mathrm{e}^{-\frac{z}{2}}z^{\frac{1}{2}+\nu}\mathrm{hypergeom}\left(\left[\frac{1}{2}+\nu-\mu\right],\left[1+2\nu\right],z\right)$
(62)

where the wave function of the free KG equation in this system given as

$\psi\left(\rho\right)=\mathcal{N}\left(\mathit{\rho^{-\frac{1}{2}}\,\mathrm{e}^{-\left(\sqrt{\xi-E^{2}}\right)\,\rho}}\left(2\left(\sqrt{\xi-E^{2}}\right)\,\rho\right){}^{\frac{1}{2}+\mathrm{i}\left(\sqrt{\kappa^{2}-\sigma^{2}}\right)}\right)R\left(\rho\right)$
(63)

and

$R\left(\rho\right)={}_{1}F_{1}\left(\left[\frac{2\,\mathrm{i}\left(\sqrt{\kappa^{2}-\sigma^{2}}\right)\,\left(\sqrt{\kappa-E^{2}}\right)+2E\kappa}{2\left(\sqrt{\xi-E^{2}}\right)}+\frac{1}{2}\right],\left[1+2\,\mathrm{i}\left(\sqrt{\kappa^{2}-\sigma^{2}}\right)\right],2\left(\sqrt{\xi-E^{2}}\right)\,\rho\right)$
(64)

using the condition of quantification

$\frac{2\,\mathrm{i}\left(\sqrt{\kappa^{2}-\sigma^{2}}\right)\,\left(\sqrt{\kappa-E^{2}}\right)+2E\kappa}{2\left(\sqrt{\xi-E^{2}}\right)}+\frac{1}{2}=-n$
(65)

After the mathematical simplification of the equation, We find the energy
phrase given in the following form:

$\begin{array}[]{c}E^{\pm}\left(n\right)=\pm\left(\frac{16\left(\sqrt{\xi^{2}\kappa^{2}\left(2n+1\right){}^{2}\left(\kappa^{2}-\sigma^{2}\right)}\right)}{16\left[n^{4}+2n^{3}+\sigma^{4}+4\kappa^{4}\right]+\left[32\sigma^{2}+24\right]n^{2}+\left[32\sigma^{2}+8\right]n+8\left[-8\kappa^{2}+1\right]\sigma^{2}+1}\right)\\\
\pm\left(\frac{\xi\left[16\left(n^{4}+2n^{3}\right)+\left(-16\left[\kappa^{2}+2\sigma^{2}\right]+24\right)n^{2}+\left(-16\left[\kappa^{2}+2\sigma^{2}\right]+8\right)n+16\sigma^{4}+8\left(-6\kappa^{2}+1\right)\sigma^{2}+4\left[8\kappa^{4}-\kappa^{2}\right]+1\right]}{16\left[n^{4}+2n^{3}+\sigma^{4}+4\kappa^{4}\right]+\left[32\sigma^{2}+24\right]n^{2}+\left[32\sigma^{2}+8\right]n+8\left[-8\kappa^{2}+1\right]\sigma^{2}+1}\right)\end{array}$
(66)

This last equation represents the energy of Klein-Gordon’s free equation as a
quantitative number $n$ where $E^{\pm}\left(n\right)=f(n)$. We note as a
result that this last energy is a function dependent of $n$ compared to the
result in the first part by the connotation of the Bessel function, which
explains the actual effect of Coulomb-type potential.

##  V Conclusion

We have been investigating KGO immersed in a non-trivial background
characterized by ingredients that characterize the presence of curvature,
torsion and extra dimension, that is, a KKT spacetime with cosmic string and
screw-like dislocation, associated with curvature and torsion, respectively.
Furthermore, the extra dimension of KKT provides a quantum flux, which
influences the quantum dynamics of KGO.

Through a purely analytical analysis, we show that the KGO is influenced not
only by curvature, torsion and the quantum flux associated with the extra
dimension, but also by the possibility of modifying the mass term of the
Klein-Gordon equation.

The influence of curvature and torsion on the KGO can be visualized through a
redefinition of the angular momentum eigenvalues, which are now dependent on
the parameters associated with the cosmic string and the screw-like
displacement. Furthermore, the quantum flux coming from the extra dimension is
part of this quantum effect of "correcting" the angular momentum eigenvalues.
These quantum effects can be seen as an effect analogous to ABE.

The modification of the mass term of the Klein-Gordon equation drastically
modifies the energy profile of the KGO, that is, the central potentials
inserted in the axial equation provide totally different relativistic energy
profiles.

…………………………………………….

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