# On The Feshbach–Villars Oscillator (FVO) Under Coulomb-Type Potential In The
Cosmic Dislocation Space-Time

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  Marwan Al-Raeei  mhdm-ra@scs-
net.org<EMAIL_ADDRESS>Faculty of Science,Damascus
University,Damascus,Syria

(August 28, 2024)

###### Abstract

In this paper, we investigate the quantum mechanical dynamics of the massive
and relativistic Feshbach-Villars oscillator in cosmic dislocation space-time
induced by a coulomb-type potential. The first-order Feshbach-Villars version
of the Klein-Gordon equation is used to find movement equations. Wave
functions and associated energy have been calculated (both in the free case
and in the interaction case).

We analyze the impact of dislocation topology on this interaction. As a
result, the effect of the dislocation on the quantum system under study is
examined.

Klein-Gordon equation, Feshbach–Villars Oscillator, topological defects,
Cosmic dislocation space-time, coulomb-type potential, Biconfluent Heun
function .

###### pacs:

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

##  I Introduction

The impact of the gravitational field on the dynamics of quantum mechanical
systems is of wide interest. On the one hand, Einstein’s theory of general
relativity (GR)(key-56, ) gives a convincing explanation of gravity as a
geometric characteristic of space-time. It proves, in particular, that the
classical gravitational field is a manifestation of space-time curvature. It
has predicted the existence of gravitational waves (key-57, ) and black holes
(key-58, ) for example.Quantum mechanics (QM) is the framework for
understanding the behavior of particles on a tiny scale (key-59, ). It is a
very effective theory (usually quantum field theory) in describing how small
particles interact and how three of nature’s four fundamental forces emerge:
weak, strong, and electromagnetic interactions (key-60, ). However, attempts
to develop a unified theory that can reconcile general relativity with quantum
mechanics, i.e. a theory of quantum gravity, have hit various roadblocks and
technical challenges that have yet to be overcome (key-6, ; key-7, ) .

One basic method for establishing a broad picture of how the gravitational
field affects relativistic particles at the quantum level is to generalize
aspects of the relativistic dynamics of particles in flat Minkowski space to
an arbitrary curved background geometry(key-8, ; key-9, ). As a result, the
approach can be adapted to deal with different models in which the concept of
curvature appears, incorporating more predictions on the values of macroscopic
observables that are required to make relevant experimental verification of
certain phenomenological consequences, particularly in astrophysics and
cosmology.Furthermore, comprehending the thermodynamic behavior of
relativistic particles where gravitational effects must be considered (key-10,
; key-11, ; key-12, ), as well as analyzing the associated features, i.e., the
fundamental statistical quantities, would provide the possibility of obtaining
useful and essential results in the context of describing the quantum behavior
of gravity.

Topological defects (domain walls, cosmic strings, monopoles, and textures)
have been extensively researched over the last few decades and continue to be
one of the most active disciplines in condensed matter physics, cosmology,
astrophysics, and elementary particle models. It is thought that these
structures arose as a result of the Kibble mechanism (key-68, ). (key-14, ;
key-15, ), where the defects emerge during the cooling of the early universe
in symmetry-breaking phase transitions (key-16, ; key-17, ). The particular
fault in question is cosmic strings (for additional information, see (key-73,
)). These items (whether static or revolving) can have discernible impacts.
They, for example, offer a potential technique for seeding galaxy formation
and gravitational lensing effects.Furthermore, by investigating cosmic strings
and their characteristics, we may learn a lot about particle physics at very
high energies in many settings. Furthermore, the potential that cosmic strings
may act like superconducting wires has been proposed in current physics, with
fascinating implications.

The harmonic oscillator (HO) has long been recognized as an important
instrument in many branches of theoretical physics (key-19, ). It is a well-
studied perfectly solvable model that may be used to examine different
difficult issues using quantum mechanics (key-74, ). Furthermore, the
relativistic extension of the quantum harmonic oscillator gives a useful model
for understanding a wide range of molecular, atomic, and nuclear interactions.
Indeed, when working with such a model, the property of having a complete set
of precise analytical solutions can give rise to radically different
interpretations of many mathematical and physical events, and hence related
applications can be obtained via the underlying formulation.

The behavior of various relativistic quantum systems is critically dependent
on the Dirac oscillator (DO). As Itô et al. (key-75, ) indicated in previous
advances of spin-1/2 particle dynamics with a linear trajectory. They
demonstrated that the system’s non-relativistic limit yields an ordinary
harmonic oscillator with a large spin-orbit coupling term. Actually, Moshinsky
and Szczepaniak (key-76, ) found that the above-mentioned DO could be derived
from the free Dirac equation by introducing an external linear potential
through a minimum replacement of the momentum operator
$\hat{p}\longrightarrow\hat{p}-im\omega\beta\hat{r}$.It is worth noting that,
in addition to the theoretical focus on researching the DO, substantial
insights may be achieved by examining physical interpretation, which is
undoubtedly important in comprehending many pertinent applications.

Inspired by DO, a similar formalism for the case of bosonic particles was
developed, and it was dubbed a Klein-Gordon oscillator (KGO) (key-23, ;
key-24, ). Several writers have lately been working on the covariant version
of this model in curved space-times and other configurations. Numerous
contributions have been made to the subject of relativistic quantum motions of
scalar and vector particles under gravitational effects produced by various
curved space-time geometries; for example, Ref. (key-79, ) studies the problem
of the interaction between KGO coupled harmonically with topological defects
in Kaluza-Klein theory. Ref. (key-80, ) investigates the relativistic quantum
dynamics of spin-0 particles in a spinning cosmic string space-time with
Coulomb-type scalar and vector potentials. Furthermore, rotating effects on
the scalar field in cosmic string space-time, space-time with space-like
dislocation, and space-time with spiral dislocation have been examined in Ref.
(key-81, ). Recently, the authors of Ref. (key-28, ) examined the KGO in a
cosmic string space-time and investigated the effects of the rotating frame
and non-commutativity in momentum space. Furthermore, in Ref. (key-82, ), the
KGO was exposed to a magnetic quantum flux in the presence of a Cornell-type
scalar and Coulomb-type vector potentials in a spinning cosmic string space-
time.

Attempts to investigate the relativistic spin-0, spin-1 bosons and spin-1/2
fermions wave functions and their time evolution have been pursued by various
authors (key-30, ; key-31, ; key-32, ) making use of the Hamiltonian form i.e,
having Schrodinger-type equations. The so-called Feshbach-Villars (FV)
equations (key-33, ) are of particular interest in this respect. These
equations were initially constructed in the purpose of permitting a
relativistic single particle interpretation of the second-order KG equation.
For the later case, FV equations originate from splitting the KG wave function
into two components in order to obtain an equation with first order time
derivative. In recent decades, a number of papers have been produced with the
aim of exploring the relativistic dynamical properties of single particles and
solving their wave equations by adopting the FV scheme (e.g,Refs.(key-34, ;
key-35, ; key-36, ; key-37, ; key-38, ; key-39, ; key-40, ) and other related
references cited therein),Bouzenada et al (key-41, ) investigate the Feshbach-
Villars oscillator (FVO) case in spinning cosmic string space-time and discuss
some findings ( thermal properties and density of this systhem).

This paper investigates the effect of a Coulomb-type potential on the Klein-
Gordon oscillator. Several publications(key-42, ; key-43, ; key-44, ; key-45,
; key-46, ) have recently considered the confinement of a relativistic scalar
particle to a Coulomb potential. The approach for inserting a scalar potential
into the Klein-Gordon equation is as follows, as explained in Ref. (key-45, ).
The electromagnetic 4-vector potential is introduced in the same way.This is
accomplished by altering the momentum operator $p_{\alpha}=i\partial_{\alpha}$
as follows:$p_{\alpha}\rightarrow p_{\alpha}-qA_{\alpha}\left(x\right)$.
Another technique was provided in Ref. (key-46, ) by changing the mass term
to:$m\rightarrow m+S(\overrightarrow{r},t)$ , where $S(\overrightarrow{r},t)$
denotes the scalar potential.This change in the mass term has been studied in
recent decades, for example, by examining the behavior of a Dirac particle in
the presence of a static scalar potential and a Coulomb potential (key-46, )
and a relativistic scalar particle in cosmic string spacetime (key-47, ). We
study the effect of a Coulomb-type potential on the Klein-Gordon oscillator in
this work by introducing the scalar potential as a modification to the mass
component in the Klein-Gordon equation.We obtain bound state solutions to the
Klein-Gordon equation for both attractive and repulsive Coulomb-type
potentials and demonstrate a quantum effect characterized by the angular
frequency of the Klein-Gordon oscillator being dependent on the quantum
numbers of the system, implying that not all angular frequency values are
allowed.

This paper is organized as follows. In the next section, we derive the FV
equations for scalar bosons in Mikowski and static Cosmic string space-time,
taking both the free and interaction cases into account. We present the KG
oscillator in Hamiltonian form, then solve the resulting equations to get the
eigenstates and energy levels. Section 3 extends the preceding equations by
considering the geometry of the cosmic dislocation space-time; the same
technique as Section 2 is applied in this section. Section 4 investigates the
Feshbach-Villars oscillator and the quantification of the energy system based
on these variables,we are using Frobenius method. Section 5 contains our
conclusions.We shall always use natural units $\hbar=c=1$ throughout the
article, and our metric convention is $(+,-,-,-)$.

##  II The FV Representation of Feshbach-Villars (Spin-0) in Minkowski Space-
time

###  II.1 An Overview of the Feshbach-Villars Approximation

This section discusses the relativistic quantum description of a spin-0
particle propagating in Minkowski space-time using the metric tensor
$\eta_{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right)$.The usual covariant KG
equation for a scalar massive particle $\Phi$ with mass $m>0$ is (key-48, ;
key-49, )

$\left(\eta^{\mu\nu}D_{\mu}D_{\nu}+m^{2}\right)\Phi(x,t)=0,$ (1)

The minimally-coupled covariant derivative is denoted by
$D_{\mu}=\left(p_{\mu}-ieA_{\mu}\right)$. The classical four momentum is
$p_{\mu}=\left(E,-p_{i}\right)$, while the electromagnetic four potential is
$A_{\mu}=\left(A_{0},-A_{i}\right)$. The magnitude of the particle charge is
given by e.

It is worth noting at this point that (1) may be expressed in Hamiltonian form
with the time first derivative, i.e. as a Schrödinger-type equation.

$\mathcal{H}\Phi\left(x,t\right)=i\,\frac{\partial}{\partial t}\Phi(x,t),$ (2)

The Hamiltonian $\mathcal{H}$ may be defined using the FV linearization
process, which involves converting 1 to a first order in time differential
equation. The two component wave function (key-46, ) is introduced. (key-50,
),

$\Phi(x,t)=\left(\begin{array}[]{c}\phi_{1}(x,t)\\\
\phi_{2}(x,t)\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1+\frac{i}{m}\mathcal{D}\\\
1-\frac{i}{m}\mathcal{D}\end{array}\right)\psi(x,t),$ (3)

Here, $\psi(x,t)$ obeys the KG wave equation, and $\mathcal{D}$ is defined in
such a way that

$\mathcal{D}=\frac{\partial}{\partial t}+ieA_{0}(x),$ (4)

The aforementioned transformation (3) involves inserting wave functions that
meet the requirements.

$\psi=\phi_{1}+\phi_{2},\qquad
i\mathcal{D}\psi=m\left(\phi_{1}-\phi_{2}\right).$ (5)

It is more convenient to write for our subsequent review,

$\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{2m}\left[m+i\frac{\partial}{\partial
t}-eA_{0}\right]\psi$ (6) $\displaystyle\phi_{2}$
$\displaystyle=\frac{1}{2m}\left[m-i\frac{\partial}{\partial
t}+eA_{0}\right]\psi,$

Eq. (1) becomes equivalent

$\displaystyle\left[i\frac{\partial}{\partial
t}-eA_{0}\right]\left(\phi_{1}+\phi_{2}\right)$
$\displaystyle=m\left(\phi_{1}-\phi_{2}\right)$ (7)
$\displaystyle\left[i\frac{\partial}{\partial
t}-eA_{0}\right]\left(\phi_{1}-\phi_{2}\right)$
$\displaystyle=\left[\frac{\left(p_{i}-eA_{i}\right)^{2}}{m}+m\right]\left(\phi_{1}+\phi_{2}\right),$

The addition and subtraction of these two equations yields a system of first
order coupled differential equations

$\displaystyle\frac{\left(p_{i}-eA_{i}\right)^{2}}{2m}\left(\phi_{1}+\phi_{2}\right)+\left(m+eA_{0}\right)\phi_{1}$
$\displaystyle=i\frac{\partial\phi_{1}}{\partial t}$ (8)
$\displaystyle\frac{-\left(p_{i}-eA_{i}\right)^{2}}{2m}\left(\phi_{1}+\phi_{2}\right)-\left(m-eA_{0}\right)\phi_{2}$
$\displaystyle=i\frac{\partial\phi_{2}}{\partial t},$

The FV Hamiltonian of a scalar particle in the presence of electromagnetic
interaction may be expressed using Eqs. (8) as

$\mathcal{H}_{K-G}=\left(\tau_{3}+i\tau_{2}\right)\frac{\left(p_{i}-eA_{i}\right)^{2}}{2m}+m\tau_{3}+eA_{0}(x),$
(9)

where $\tau_{i}\,\left(i=1,2,3\right)$ are the conventional $2\times 2$ Pauli
matrices given by

$\tau_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\tau_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\tau_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (10)

It’s worth noting that the Hamiltonian (9) meets the generalized hermicity
requirement (If there is an invertible, Hermitian, linear operator $\beta$
such that $\mathcal{H}^{\dagger}=\beta\mathcal{H}\beta^{-1}$, the Hamiltonian
$\mathcal{H}$ is said to be pseudo-Hermitian. (key-51, )).

$\mathcal{H}_{K-G}=\tau_{3}\left(\mathcal{H}_{K-G}^{\dagger}\right)\tau_{3},\qquad\mathcal{H}_{K-G}^{\dagger}=\tau_{3}\left(\mathcal{H}_{K-G}\right)\tau_{3}.$
(11)

The one dimensional FV Hamiltonian reduces to for free particle propagation,
i.e., no interaction is assumed left $\left(A_{\mu}=0\right)$.

$\mathcal{H}_{0}=\left(\tau_{3}+i\tau_{2}\right)\frac{p_{x}^{2}}{2m}+m\tau_{3},$
(12)

The solutions to the time-independent free Hamiltonian are simply stationary
states. Assuming the solution (key-36, ),

$\Phi\left(x,t\right)=\Phi\left(x\right)e^{-iEt}=\left(\begin{array}[]{c}\phi_{1}\left(x\right)\\\
\phi_{2}\left(x\right)\end{array}\right)e^{-iEt},$ (13)

with E denoting the system’s energy. As a result, Eq. (2) may be represented
as

$\mathcal{H}_{0}\Phi\left(x\right)=E\Phi\left(x\right),$ (14)

This is the one-dimensional FV equation of the free relativistic spin-0
particle, and it is performed in order to have an alternate Schrödinger-type
to KG equation. In what follows, the aforesaid approach will be utilized to
determine the dislocation solutions to wave equations in curved space-time.

##  III The FV Representation of Spin-0 Particle in Cosmic Dislocation Space-
time

The goal of this part is to investigate the KGO in the backdrop geometry of a
cosmic string using the FV technique. It is widely known that the generally
covariant relativistic wave equations of a scalar particle in a Riemannian
space-time characterized by the metric tensor $g_{\mu\nu}$ may be found by
reformulating the KG equation so that( see, e.g, the textbooks(key-8, ; key-9,
) )

$\left(\square+m^{2}-\mathcal{\xi}R\right)\Phi(x,t)=0,$ (15)

where $\square$ is the Laplace-Beltrami operator denoted by

$\square=g^{\mu\nu}D_{\mu}D_{\nu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right),$
(16)

$\xi$ denotes a real dimensionless coupling constant, and R is the Ricci
scalar curvature given by $R=g^{\mu\nu}R_{\mu\nu}$, where $R_{\mu\nu}$ is the
Ricci curvature tensor. The inverse metric tensor is $g^{\mu\nu}$, and
$g=\det\left(g^{\mu\nu}\right)$.

We would now want to investigate the quantum dynamics of spin-0 particles in
the space-time caused by a (3+1)-dimensional dislocation, as well as develop
the relevant FV formulation.

###  III.1 Feshbach-Villars oscillator in the cosmic dislocation space-time

Before we study the KGO in the Hamiltonian representation, let us first derive
the KG wave equation for the free relativistic scalar particle propagating in
the cosmic string space-time that is assumed to be static and cylindrically
symmetric.

In this part, we will discuss the topological defect that serves as the
foundation for our study. Inspired by the description of an edge dislocation
in crystalline solids, we build a generalization of this topological defect in
gravity. We can observe that an edge dislocation is a spiral dislocation,
which is a deformation of a circle into a spiral (key-52, ). The line element
describing the space-time backdrop with this topological defect is (using the
units $\hbar=c=1$) (key-53, ; key-54, ; key-55, )

$\displaystyle ds^{2}$ $\displaystyle=g_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=dt^{2}-dr^{2}-\left(\alpha^{2}r^{2}+\chi^{2}\right)d\varphi^{2}+2\chi
dzd\varphi-dz,$ (17)

where $\chi$ is a constant value relating to the defect’s distortion. By
$\chi=\frac{\left|\overrightarrow{b}\right|}{2\pi}$, the parameter is also
connected to the Burgers vector $\overrightarrow{b}$. Here $-\infty\leq
t\leq+\infty$, $r\geq 0$, $0\leq\varphi\leq 2\pi$, $-\infty\leq
z\leq+\infty$,and $\alpha\in[0,1[\>$: is the angular parameter that defines
the angular deficit $\delta\varphi=2\pi(1-\alpha)$, which is connected to the
string’s linear mass density mu by $\alpha=1-4\mu$(It should be noted that
this metric provides an accurate solution to Einstein’s field equations for
$0\leq\mu<1/4,$, and that by setting $\varphi^{\prime}=\alpha\varphi$, it
represents a flat conical outer space with angle deficit
$\delta\phi=8\pi\mu$).

When the metric and inverse metric tensor components are, respectively,

$g_{\mu\nu}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\
0&0&-\left(\alpha^{2}r^{2}+\chi^{2}\right)&\chi\\\
0&0&\chi&-1\end{array}\right),\quad
g^{\mu\nu}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\
0&0&-\frac{1}{\alpha^{2}r^{2}}&-\frac{\chi}{\alpha^{2}r^{2}}\\\
0&0&-\frac{\chi}{\alpha^{2}r^{2}}&-\left(1+\frac{\chi^{2}}{\alpha^{2}r^{2}}\right)\end{array}\right)$
(18)

It is worth noting that the topic of spinless heavy particles in the geometry
formed by a cosmic dislocation background has been studied in a number of
studies (for example,(key-56, ; key-57, )).

To obtain the FV form of the KG wave equation in curved manifolds, we will use
the approach provided in references (key-58, ; key-59, ). The generalized
Feshbach-Villars transformation is used (GFVT) An identical transformation for
characterizing both large and massless particles was presented earlier in
Ref.(key-60, ). The components of the wave function $\Phi$ in the GFVT are
provided by (key-58, ).

$\psi=\phi_{1}+\phi_{2},\qquad
i\tilde{\mathcal{D}}\psi=\mathcal{N}\left(\phi_{1}-\phi_{2}\right),$ (19)

where $\mathcal{N}$ is an arbitrary nonzero real parameter and
$\tilde{\mathcal{D}}=\frac{\partial}{\partial t}+\mathcal{Y},$ is specified,
with

$\mathcal{Y}=\frac{1}{2}\left\\{\partial_{i},\sqrt{-g}\frac{g^{0i}}{g^{00}}\right\\},$
(20)

The anti-commutator is denoted by the curly bracket in Eq. (20). The
Hamiltonian for the aforementioned transformation is

$\mathcal{H}_{GFVT}=\tau_{z}\left(\frac{\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)+i\tau_{y}\left(\frac{-\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)-i\mathcal{Y},$
(21)

with

$\displaystyle\mathcal{T}$
$\displaystyle=\partial_{i}\frac{G^{ij}}{g^{00}}\partial_{j}+\frac{m^{2}-\xi
R}{g^{00}}+\frac{1}{\mathcal{F}}\nabla_{i}\left(\sqrt{-g}G^{ij}\right)\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\sqrt{\frac{\sqrt{-g}}{g^{00}}}G^{ij}\nabla_{i}\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\frac{1}{4\mathcal{F}^{4}}\left[\nabla_{i}\left(\mathcal{U}^{i}\right)\right]^{2}$
$\displaystyle\qquad-\frac{1}{2\mathcal{F}^{2}}\nabla_{i}\left(\frac{g^{0i}}{g^{00}}\right)\nabla_{j}\left(\mathcal{U}^{i}\right)-\frac{g^{0i}}{2g^{00}\mathcal{F}^{2}}\nabla_{i}\nabla_{j}\left(\mathcal{U}^{i}\right),$
(22)

where

$G^{ij}=g^{ij}-\frac{g^{0i}g^{0j}}{g^{00}},\qquad\mathcal{F}=\sqrt{g^{00}\sqrt{-g}},\qquad\mathcal{U}^{i}=\sqrt{-g}g^{0i}.$
(23)

Here and everywhere $(i,j=1,2,3)$. We see that the initial FV transformations
are fulfilled for $\mathcal{N}=m$.

Now, using the metric 25, it is simple to see that $R=0$, implying that space-
time is locally flat (no local gravity), and so the coupling component is
vanishing. The condition $\xi=0$ is known as minimum coupling. However, in
massless theory, $\xi$ equals 1/6. (in 4 dimensions). The equations of motion
are then conformally invariant in this later instance.

A simple computation yields $\mathcal{Y}=0,$ and we then obtain

$\mathcal{T}=-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}-\frac{1}{\alpha^{2}r^{2}}\frac{d^{2}}{d\varphi^{2}}-\left(1+\frac{\chi^{2}}{\alpha^{2}r^{2}}\right)\frac{d^{2}}{dz^{2}}-\frac{2\chi}{\alpha^{2}r^{2}}\left(\frac{d}{d\varphi}\frac{d}{dz}\right)+m^{2},$
(24)

Using these techniques to obtain the Hamiltonian 21, one may suppose a
solution of the type we want cylindrically symmetric, i.e. solutions with
rotational symmetry in the (x,y)-plane, due to the temporal and angular
independence in the metric 25.

$\Phi(t,r,\varphi,z)=\Phi(r)e^{-i\left(Et-j\varphi-Kz\right)},$ (25)

where $j=0,\pm 1,\pm 2,..$ are the eigenvalues of the $z$-component of the
angular momentum operator. The KG equation 15 may be written equivalently to
the following two coupled equations

$\displaystyle\left(\mathcal{N}^{2}+\mathcal{T}\right)\phi_{1}+\left(-\mathcal{N}^{2}+\mathcal{T}\right)\phi_{2}$
$\displaystyle=2\mathcal{N}E\phi_{1}$
$\displaystyle-\left(\mathcal{N}^{2}+\mathcal{T}\right)\phi_{2}-\left(-\mathcal{N}^{2}+\mathcal{T}\right)\phi_{1}$
$\displaystyle=2\mathcal{N}E\phi_{2},$ (26)

The sum and difference of the two previous equations yields a second order
differential equation for the field $\psi$. As a result, the radial equation
is as follows:

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\zeta^{2}}{r^{2}}+\kappa\right]\psi\left(r\right)=0,$
(27)

where we have set

$\zeta=\frac{j+\chi K}{\alpha},\qquad\kappa=\sqrt{E^{2}-m^{2}-K^{2}}.$ (28)

We can observe that Eq. (27) is a Bessel equation and its general solution is
defined by (key-61, )

$\psi\left(r\right)=A\,J_{|\zeta|}\left(\kappa
r\right)+B\,Y_{|\zeta|}\left(\kappa r\right),$ (29)

where $J_{|\zeta|}\left(\kappa r\right)$ and $Y_{|\zeta|}\left(\kappa
r\right)$ are the Bessel functions of order $\zeta$ and of the first and the
second kind, respectivement. Here $A$ and $B$ are arbitrary constants. We
notice that at the origin when $\zeta=0$, the function
$J_{|\zeta|}\left(\kappa r\right)\neq 0$. However, $Y_{|\zeta|}\left(\kappa
r\right)$ is always divergent at the origin. In this case, we will consider
only $J_{|\zeta|}\left(\kappa r\right)$ when $\zeta\neq 0$. Hence, we write
the solution to Eq. (27) as follows

$\psi\left(r\right)=A\,J_{\frac{|j+\chi
K|}{\alpha}}\left(\sqrt{E^{2}-m^{2}-K^{2}}\,r\right),$ (30)

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

$\psi\left(t,r,\varphi,z\right)=\left|\mathcal{C}_{1}\right|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\varphi-
Kz\right)}\,J_{\frac{|j+\chi
K|}{\alpha}}\left(\sqrt{E^{2}-m^{2}-K^{2}}\,r\right),$ (31)

The constant $\left|\mathcal{C}_{1}\right|$ can be obtained by applying the
appropriate normalization condition to the KG equation (e.g., see Ref.(key-62,
; key-63, )), but it is fortunate that failing to determine the normalization
constants throughout this manuscript has no effect on the final results.

Now we’ll look at the specific instance where we wish to extend the GFVT for
the KGO. In general, we must substitute the momentum operator in Eq. (15). As
a result, Eq. (30) may be rewritten as follows.

$\mathcal{T}=\frac{1}{\sqrt{-g}}\left(\frac{\partial}{\partial r}-m\omega
r\right)\left(-\sqrt{-g}\right)\left(\frac{\partial}{\partial r}+m\omega
r\right)-\frac{1}{\alpha^{2}r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}-\left(1+\frac{\chi^{2}}{\alpha^{2}r^{2}}\right)\frac{d^{2}}{dz^{2}}-\frac{2\chi}{\alpha^{2}r^{2}}\left(\frac{d}{d\varphi}\frac{d}{dz}\right)+m^{2},$
(32)

Similarly, the following differential equation may be obtained using a simple
calculation based on the approach described above.

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}+m^{2}\omega^{2}r^{2}-\frac{\sigma^{2}}{r^{2}}+\delta\right]\psi\left(r\right)=0,$
(33)

with

$\sigma^{2}=\left(\frac{j+\chi
K}{\alpha}\right)^{2},\qquad\delta=E^{2}-m^{2}-K^{2}+2m\omega.$ (34)

The KGO for a spin-0 particle in the space-time of a static cosmic string is
given by Eq. (33). To derive the solution to this problem, we first suggest a
radial coordinate transformation.

$\mathcal{Q}=m\omega r^{2},$ (35)

subsitutuing the expression for $\chi$ into Eq. (33), we obtain

$\left[\frac{d^{2}}{d\mathcal{Q}^{2}}+\frac{1}{\mathcal{Q}}\frac{\partial}{d\mathcal{Q}}-\frac{\sigma^{2}}{4\mathcal{Q}^{2}}+\frac{\delta}{4m\omega\mathcal{Q}}-\frac{1}{4}\right]\psi\left(\chi\right)=0.$
(36)

So, if we look at the asymptotic behavior of the wave function at the origin
and infinity, and we’re looking for regular solutions, we may assume a
solution of the type

$\psi\left(\mathcal{Q}\right)=\mathcal{Q}^{\frac{\left|\sigma\right|}{2}}e^{-\frac{\mathcal{Q}}{2}}F\left(\mathcal{Q}\right),$
(37)

As previously, we can plug this back into Eq. (36), and we get

$\mathcal{Q}\frac{d^{2}F\left(\mathcal{Q}\right)}{d\mathcal{Q}^{2}}+\left(|\sigma|+1-\chi\right)\frac{dF\left(\mathcal{Q}\right)}{d\mathcal{Q}}-\left(\frac{|\sigma|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}\right)F\left(\mathcal{Q}\right)=0,$
(38)

This is the confluent hypergeometric equation (key-64, ), the solutions to
which are defined in terms of the kind of confluent hypergeometric function.

$F\left(\mathcal{Q}\right)=_{1}F_{1}\left(\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},+1,\mathcal{Q}\right),$
(39)

We should note that the solution (39) must be a polynomial function of degree
$n$. However, taking $n\rightarrow\infty$ imposes a divergence issue. We can
have a finite polynomial only if the factor of the last term in Eq. (38) is a
negative integer, meaning,

$\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}=-n\qquad,n=0,1,2,\cdots.$
(40)

With this result and the parameters (34), we may derive the quantized energy
spectrum of KGO in the cosmic dislocation space-time, and hence,

$E\left(n,\alpha,j\right)=\pm\sqrt{4m\omega n+\frac{2m\omega\left|j+\chi
K\right|}{\alpha}+m^{2}+K^{2}},$ (41)

We may notice that the energy relies clearly on the angular deficit $\alpha$.
In other words, because to the presence of the wedge angle, the curvature of
space-time that is impacted by the topological defect, i.e., the cosmic
string, would affect the relativistic dynamics of the scalar particle by
creating a gravitational field.

The corresponding wave function is given by

$\psi\left(t,r,\varphi,z\right)=\left|\mathcal{C}_{2}\right|\left(m\omega
r^{2}\right)^{\frac{\left|j+\chi K\right|}{2\alpha}}e^{-\frac{m\omega
r^{2}}{2}}{}_{1}F_{1}\left(\frac{\left|j+\chi
K\right|}{2\alpha}-\frac{\delta}{4m\omega}+\frac{1}{2},\frac{\left|j+\chi
K\right|}{\alpha}+1,m\omega r^{2}\right),$ (42)

Thereafter, the general eigenfunctions are written as

$\psi\left(t,r,\varphi,z\right)=\left|\mathcal{C}_{2}\right|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)\left(m\omega
r^{2}\right)^{\frac{\left|j+\chi K\right|}{2\alpha}}e^{-\frac{m\omega
r^{2}}{2}}e^{-i\left(Et-j\varphi-Kz\right)}{}_{1}F_{1}\left(\frac{\left|j+\chi
K\right|}{2\alpha}-\frac{\delta}{4m\omega}+\frac{1}{2},\frac{\left|j+\chi
K\right|}{\alpha}+1,m\omega r^{2}\right),$ (43)

where $\left|\mathcal{C}_{2}\right|$ is the normalization constant.

###  III.2 In the cosmic dislocation space-time, the Feshbach-Villars
oscillator Coulomb-Type Potentials

In this part, we will look at the KGO in the context of a cosmic dislocation.
The equations of motion of a scalar particle can be obtained by examining the
GFVT, as in the instance examined in Sec.LABEL:sec:3. Numerous writers
investigated the quantum dynamics of relativistic particles in the space-time
of a cosmicdislocation, and a variety of models were addressed. Mazur, for
example, explored the quantum mechanical features of heavy (or massless)
particles in the gravitational field of a spinning cosmic dislocation in a
prior publication (key-65, ). He demonstrated that energy should be quantized
when the string has non-zero rotational momentum.Subsequently, Gerbert and
Jackiw (key-66, ) showed solutions to the KG and Dirac equations in the
(2+1)-dimensional space-time produced by a large point particle with arbitrary
angular momentum. The vacuum expectation value of the stress-energy tensor for
a massless scalar field conformally related to gravity was explored in Ref.
(key-67, ). The authors of (key-66, ) investigated the behavior of a quantum
test particle meeting the Klein-Gordon equation in a spinning cosmic string’s
space-time.

Additionally, it was demonstrated in Ref. (key-68, ) that the extrema of the
field’s energy for specified angular and linear momenta may be defined as
spinning cosmic string solutions of $U(1)$ scalar field theory with a
cylindrically symmetric energy density. Furthermore, in Ref. (key-69, ),
topological and geometrical phases owing to the gravitational field of a
cosmic string with mass and angular momentum were explored.

The gravitational effects of rotating cosmic strings have recently piqued the
curiosity of researchers studying the dynamics and characteristics of
relativistic quantum particles. Vacuum fluctuations for a massless scalar
field surrounding a spinning cosmic string, for example, were examined using a
renormalization approach in Ref. (key-70, ). Similarly, in Ref. (key-71, ),
the vacuum polarization of a scalar field in the gravitational backdrop of a
spinning cosmic thread was examined. Additionally, the authors of Ref.
(key-72, ) used a completely relativistic technique to investigate the Landau
levels of a spinless heavy particle in the spacetime of a spinning cosmic
string.

Wang et al. (key-73, ) studied the KGO linked to a homogeneous magnetic field
in the backdrop of a revolving cosmic string. In addition, Ref.(key-74, )
addressed the problem of a spinless relativistic particle subjected to a
uniform magnetic field in the spinning cosmic string space-time. Furthermore,
in Ref. (key-75, ), the relativistic quantum dynamics of a KG scalar field
exposed to a Cornell potential in spinning cosmic string space-time were
reported. In addition, in Ref. (key-76, ), the relativistic scalar charged
particle in a spinning cosmic string space-time with Cornell-type potential
and Aharonov-Bohm effect was studied.

The topic examined in Sec. LABEL:sec:3 is extended to a more generic space-
time with non-zero angular momentum in the following discussion. In this
paper, we investigate a heavy, relativistic spin-0 particle whose wave-
function is represented by Psi and satisfies the KG equation 15 in the space-
time caused by a (3+1)-dimensional stationary cosmic dislocation described in
Coulomb-Type
Potentials$\left(S\left(r\right)=\frac{\lambda}{r}=\pm\frac{\left|\lambda\right|}{r}\right)$.

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\left(\frac{j+\chi
K}{\alpha}\right)^{2}}{r^{2}}+\left(E-\frac{\lambda}{r}\right)^{2}-m^{2}-K^{2}\right]\varphi\left(r\right)=0,$
(44)

where

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\left(\frac{\left(\frac{j+\chi
K}{\alpha}\right)^{2}-\lambda^{2}}{r^{2}}\right)-\frac{2\lambda
E}{r}+E-m^{2}-K^{2}\right]\varphi\left(r\right)=0,$ (45)

After simple algebraic manipulations we arrive at the following second order
differential equation for the radial function $\varphi(r)$

$\varphi\left(r\right)=\left|\mathcal{C}_{3}\right|r^{-\frac{1}{2}}\mathrm{WhittakerM}(-\frac{E\lambda}{\sqrt{-E^{2}+K^{2}+m^{2}}},\frac{-\alpha^{2}\lambda^{2}+(K\chi+j)^{2}}{\alpha^{2}},2\sqrt{-E^{2}+K^{2}+m^{2}}\,r),$
(46)

By simplifying the relationship between Whitakar and confluent hypergeometric
function(key-61, ), yields

$\varphi\left(r\right)=\left|\mathcal{C}_{3}\right|r^{-\frac{1}{2}}\mathrm{e}^{-\left(\sqrt{-E^{2}+K^{2}+m^{2}}\right)r}\left(2r\left(\sqrt{-E^{2}+K^{2}+m^{2}}\right)\right)^{{}^{\frac{(-2\lambda^{2}+1)\alpha^{2}+2(K\chi+j)^{2}}{2\alpha^{2}}}}F\left(r\right)$
(47)

where

$F\left(r\right)=1F1\left(\frac{\left\\{\left(-\lambda^{2}+\frac{1}{2}\right)\alpha^{2}+\left(K\chi+j)^{2}\right)\right\\}\sqrt{-E^{2}+K^{2}+m^{2}}+E\lambda\alpha^{2}}{\left(\sqrt{-E^{2}+K^{2}+m^{2}}\right)\alpha^{2}},\frac{\left(-2\lambda^{2}+1\right)\alpha^{2}+2\left(K\chi+j\right){}^{2}}{\alpha^{2}},2r\left(\sqrt{-E^{2}+K^{2}+m^{2}}\right)\right)$
(48)

We should note that the solution must be a polynomial function of degree $n$.
However, taking $n\rightarrow\infty$ imposes a divergence issue. We can have a
finite polynomial only if the factor of the last term in Eq. is a negative
integer, meaning,

$\frac{\left[\left(-\lambda^{2}+\frac{1}{2}\right)\alpha^{2}+\left(K\chi+j)^{2}\right)\right]\sqrt{-E^{2}+K^{2}+m^{2}}+E\lambda\alpha^{2}}{\sqrt{-E^{2}+K^{2}+m^{2}}\alpha^{2}}=-n\qquad,n=0,1,2,\cdots.$
(49)

With this result and the parameters (34), we may derive the quantized energy
spectrum of KGO in the cosmic dislocation space-time, and hence,

$E(n)=\pm\left(\frac{2\left(\left(2\lambda^{2}+2n+1\right)\alpha^{2}+2(K\chi+j)^{2}\right)\sqrt{\left[\left(\lambda^{4}+2\left(n+1\right)\lambda^{2}+\left(n+\frac{1}{2}\right){}^{2}\right)\alpha^{4}+2\left(K\chi+j\right){}^{2}\left(\lambda^{2}+n+\frac{1}{2}\right)\alpha^{2}+\left(K\chi+j\right){}^{4}\right]\left(K^{2}+m^{2}\right)}}{\left[4\lambda^{4}+8\left(n+1\right)\lambda^{2}+\left(2n+1\right){}^{2}\alpha^{4}+8\left(K\chi+j\right){}^{2}\left(\lambda^{2}+n+\frac{1}{2}\right)\alpha^{2}+4\left(K\chi+j\right)\right]{}^{4}}\right),$
(50)

where $\left|\mathcal{C}_{3}\right|$ is an integration constant. The complete
eigenstates are given by

$\psi\left(t,r,\varphi,z\right)=\left|\mathcal{C}_{3}\right|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)r^{-\frac{1}{2}}\mathrm{e}^{-\left(\sqrt{-E^{2}+K^{2}+m^{2}}\right)r}\left(2r\left(\sqrt{-E^{2}+K^{2}+m^{2}}\right)\right)^{{}^{\frac{(-2\lambda^{2}+1)\alpha^{2}+2(K\chi+j)^{2}}{2\alpha^{2}}}}F\left(r\right)$
(51)

From now on we proceed to study the Feshbach–Villars Oscillator in a rotating
cosmic string space-time . Firstly, we start by considering a scalar quantum
particle embedded in the background gravitational field of the space-time
described by the metric (42). In this way, we shall introduce a replacement of
the momentum operator $p_{i}\longrightarrow p_{i}+im\omega x_{i}$ where
$p_{i}=i\nabla_{i}$ in Eq. (57). Then, we have

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-m^{2}\omega^{2}r^{2}-\left(\frac{\left(\frac{j+\chi
K}{\alpha}\right)^{2}-\lambda^{2}}{r^{2}}\right)-\frac{2\lambda
E}{r}+E^{2}-m^{2}-K^{2}+2mw\right]\varphi\left(r\right)=0,$ (52)

Based on the prior studies, we will apply the GFVT to the case of KGO in the
relevant space using the same techniques as previously. Inserting Eqs. (52)
and (45) into the Hamiltonian (56), then assuming the solution (48), yields
two linked differential equations comparable to Eq.(56), but with different
values of $\mathcal{T}^{\prime}$.

Manipulating exactly the same steps before, we obtain the following radial
equation

$\left[\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-m^{2}\omega^{2}r^{2}-\frac{\vartheta^{2}}{r^{2}}-\frac{2\lambda
E}{r}+\delta\right]\varphi\left(r\right)=0,$ (53)

where we have defined

$\vartheta^{2}=\left(\frac{aE+j}{\alpha}\right)^{2}-\lambda^{2},\qquad\beta^{2}=E^{2}-m^{2}-K^{2}+2mw.$
(54)

Let us now conside $\mathcal{K}=\sqrt{m\omega}r$, and therefore rewrite the
radial equation ((53)) as follows
($\varphi\left(r\right)\rightarrow\varphi\left(\mathcal{\mathcal{K}}\right)$):

$\left[\frac{d^{2}}{d\mathcal{\mathcal{K}}^{2}}+\frac{1}{\mathcal{\mathcal{K}}}\frac{d}{d\mathcal{\mathcal{K}}}-\frac{\vartheta^{2}}{\mathcal{K}^{2}}-\frac{\delta}{\mathcal{K}}-\mathcal{K}^{2}+\frac{\beta^{2}}{m\omega}\right]\varphi\left(\mathcal{\mathcal{K}}\right)=0,$
(55)

where we have defined a new parameter

$\delta=2\lambda\sqrt{\frac{m}{\omega}},$ (56)

Let us look at the asymptotic behavior of the solutions to Eq. ((55)), which
are found for $\mathcal{\mathcal{K}}\rightarrow 0$
and$\mathcal{\mathcal{K}}\rightarrow\infty$. The behavior of the potential
solutions to Eq. ((55)) at $\mathcal{\mathcal{K}}\rightarrow 0$ and allows us
to define the function $R(\mathcal{K})$ in terms of an unknown function
$\mathcal{O}(\mathcal{K})$ as follows from Refs. [44-46]:

$\varphi(\mathcal{K})=\mathcal{K}^{\left|\gamma\right|}e^{-\frac{\mathcal{K}^{2}}{2}}\mathcal{O}\left(\mathcal{K}\right),$
(57)

Hence, by plugging the radial wave function from Eq. (11) into Eq. (9), we get

$\frac{d^{2}\mathcal{O}\left(\mathcal{K}\right)}{d\mathcal{\mathcal{K}}^{2}}+\left[\frac{\left(2\left|\gamma\right|+1\right)}{\mathcal{\mathcal{K}}}-2\mathcal{K}\right]\frac{d\mathcal{O}\left(\mathcal{K}\right)}{d\mathcal{\mathcal{K}}}+\left[\frac{\beta^{2}}{m\omega}-2\left(2\left|\gamma\right|+1\right)-\frac{\delta}{\mathcal{K}}\right]\mathcal{O}\left(\mathcal{K}\right)=0,$
(58)

The Heun biconfluent equation [44, 48-50] relates to the second order
differential equation ((58)), and the function
$\mathcal{O}\left(\mathcal{K}\right)$ is the Heun biconfluent function.

$\mathcal{O}\left(\mathcal{K}\right)=HeunB\left(2\left|\gamma\right|,0,\frac{\beta^{2}}{m\omega},2\delta,\mathcal{K}\right)$
(59)

To continue our discussion of bound state solutions, let us employ the
Frobenius method(key-81, ; key-84, ). As a result, the solution to Equation
((59)) may be expressed as a power series expansion around the origin:

$\mathcal{O}\left(\mathcal{K}\right)=\stackrel{{\scriptstyle[}}{{j}}=0]{\infty}{\sum}a_{j}\mathcal{K}^{j}$
(60)

We find the following recurrence connection by substituting the series ((60))
into ((59)):

$a_{j+2}=\frac{\delta}{\left(j+2\right)\left(j+1+\zeta\right)}a_{j+1}-\frac{\left(\varTheta-2j\right)}{\left(j+2\right)\left(j+1+\zeta\right)}a_{j}$
(61)

where $\zeta=2\left|\gamma\right|+1$ and
$\varTheta=\frac{\beta^{2}}{m\omega}-2\left(\left|\gamma\right|+1\right)$. We
may determine the additional coefficients of the power series expansion by
starting with $a_{0}=1$ and applying the relation ((60)). ((58)). As an
example,

$a_{1}=\frac{\delta}{\zeta}=2\sqrt{\frac{m}{\omega}}\lambda\left(\frac{1}{2\left|\gamma\right|+1}\right)$
(62) $\displaystyle a_{2}=$
$\displaystyle\frac{\delta^{2}}{2\zeta\left(1+\zeta\right)}-\frac{\varTheta}{2\left(1+\zeta\right)}$
$\displaystyle=$
$\displaystyle\left(\frac{m}{\omega}\right)\lambda^{2}\frac{1}{\left(2\left|\gamma\right|+1\right)\left(\left|\gamma\right|+1\right)}-\frac{\varTheta}{4\left(\left|\gamma\right|+1\right)}$
(63)

The wave function must be normalizable, as is widely known in quantum theory.
As a result, we suppose that the function
$\mathcal{O}\left(\mathcal{K}\right)$ disappears at
$\mathcal{\mathcal{K}}\rightarrow 0$ and
$\mathcal{\mathcal{K}}\rightarrow\infty$. This indicates that we have a finite
wave function everywhere, which means that there is no divergence of the wave
function at $\mathcal{\mathcal{K}}\rightarrow 0$ and
$\mathcal{\mathcal{K}}\rightarrow\infty$ thus bound state solutions may be
produced.

In Eq (58), we have, however, expressed the function
$\mathcal{O}\left(\mathcal{K}\right)$ as a power series expansion around the
origin (64). By demanding that the power series expansion (60) or the Heun
biconfluent series becomes a polynomial of degree $n$, bound state solutions
can be obtained. As a result, we ensure that
$\mathcal{O}\left(\mathcal{K}\right)$ behaves as
$\mathcal{K^{\left|\gamma\right|}}$ at $\mathcal{K}\rightarrow 0$ the origin
and disappears at(key-77, ; key-78, ) . We can see from the recurrence
relation (63) that by applying two constraints (key-76, ; key-77, ; key-78, ;
key-79, ; key-80, ; key-81, ; key-82, ; key-83, ), the power series expansion
(60) becomes a polynomial of degree $n$:

$\varTheta=2n\,\,and\,\,a_{n+1}=0$ (64)

where $n=1,2,3,....$ From the condition $\varTheta=2n$, we can obtain:

$E^{\pm}\left(n\right)=\pm\sqrt{2m\left(n+\left|\gamma\right|\right)+m^{2}+K^{2}}$
(65)

The corresponding wave function is given by

$\varphi\left(r\right)=\left|\mathcal{C}_{4}\right|\mathcal{K}^{\left|\gamma\right|}e^{-\frac{\mathcal{K}^{2}}{2}}HeunB\left(2\left|\gamma\right|,0,\frac{\beta^{2}}{m\omega},2\delta,\mathcal{K}\right)$
(66)

the final expression of the wave-function of the spinless FVO propagating in
the dislocation cosmic background can be represented as

$\psi\left(t,r,\varphi,z\right)=\left|\mathcal{C}_{4}\right|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\varphi-
Kz\right)}\mathcal{K}^{\left|\gamma\right|}e^{-\frac{\mathcal{K}^{2}}{2}}HeunB\left(2\left|\gamma\right|,0,\frac{\beta^{2}}{m\omega},2\delta,\mathcal{K}\right),$
(67)

where the parameters $\vartheta$ and $\delta$ are defined in Eq.(54).

##  IV Conclusion

The goal of this work is to investigate the relativistic dynamics of spinless
quantum particles using the Feshbach-Villars representation of two models,
namely the interaction of KGO with the gravitational field created by the
background geometry of: a) cosmic dislocation and b) cosmic dislocation with
Coulomb-Type Potential. We generated the equivalent formulations in two
distinct curved manifolds by modifying the FV formulation of scalar fields in
Minkowski space-time.

We obtained the exact solutions of both systems and we presented the quantized
energy spectra which depend on the parameters that characterize the space-time
topology.

It is not surprising that the wave-functions of our quantum system are
expressed in terms of the confluent hypergeometric functions for the free
Feshbach-Villars equation in cosmic dilocation space-time, because the former
can be described throughout the latter by using the appropriate coordinate
transformation.

It is worth noting that the Feshbach-Villars oscillator has been presented
under Coulomb-type potential. Nevertheless, as explained in Ref. (key-46, ),
the electromagnetic 4-vector potential may be introduced using the same
technique by altering the momentum operator as $p_{\alpha}\rightarrow
p_{\alpha}-qA_{\alpha}\left(x\right)$. New and fascinating results linked with
the Feshbach-Villars oscillator can be achieved by dealing with a Coulomb-type
potential via a minimum coupling. Moreover, the wave-functions of our quantum
system are represented in terms of the Biconfluent Heun functions. In this
section, we solve the differential equation using the power series approach.

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