#  Applications of the Klein-Gordon equation in the Feshbach-Villars
representation in the non-inertial cosmic string space-time

Abdelmalek Bouzenada<EMAIL_ADDRESS><EMAIL_ADDRESS>Laboratory of theoretical and applied
Physics,
Echahid Cheikh Larbi Tebessi University, Algeria  Abdelmalek Boumali
<EMAIL_ADDRESS>Laboratory of theoretical and applied Physics,
Echahid Cheikh Larbi Tebessi University, Algeria  Edilberto O. Silva
<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal do Maranhão, 65085-580 São Luís, Maranhão, Brazil

###### Abstract

We study the relativistic quantum motion of a spineless particle using the
Feshbach-Villars (FV) formalism in the spinning cosmic string spacetime. The
movement equations are derived using the first-order FV formulation of the
Klein-Gordon (KG) equation. We apply the equation of motion (a) to study the
motion of the particle confined to a rigid-wall potential, (b) motion in the
presence of a Coulomb-type potential, and (c) particle interacting with the
Feshbach-Villars oscillator (FVO). The energy levels and wave functions are
obtained for the three cases. Our study focused on the impact of rotation and
curvature on the energy levels of the particle.

Feshbach–Villars representation, cosmic string, rotating frame, Klein-Gordon
equation

###### 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 study of how the gravitational field affects the dynamics of quantum
mechanical systems holds significant importance. On the one hand, the theory
of general relativity (GR), as proposed by Einstein (key-1, 1), provides a
compelling explanation of gravity as a geometric property of space-time. It
establishes that the classical gravitational field manifests space-time
curvature and has successfully predicted phenomena such as gravitational waves
(key-2, 2) and black holes (key-3, 3). On the other hand, quantum mechanics
(QM) serves as the framework for comprehending the behavior of particles at
the microscopic level (key-4, ). Quantum mechanics, often manifested through
quantum field theory, has proven highly successful in elucidating the
interactions among small particles and the emergence of three fundamental
forces: weak, strong, and electromagnetic interactions (key-5, ). However, the
endeavor to construct a unified theory that reconciles general relativity with
quantum mechanics, known as quantum gravity, has encountered numerous
obstacles and unresolved technical challenges (key-6, ; key-7, ).

A key approach to developing a theory that bridges the gap between gravity and
relativistic quantum mechanics is by generalizing certain aspects of particle
dynamics in flat Minkowski space to curved geometries (key-8, ; key-9, ). This
allows for the extension of this formulation, providing a comprehensive
understanding of how the gravitational field influences the quantum behavior
of relativistic particles. By incorporating arbitrarily curved backdrops, a
broader framework can be constructed, shedding light on the intricate
interplay between gravity and quantum mechanics at the relativistic level.

The versatility of this technique allows for its adaptation to address a wide
array of models involving the concept of curvature, enabling the integration
of a broader set of predictions concerning the values of macroscopic
observables. These predictions play a crucial role in facilitating meaningful
experimental verification of various phenomenological consequences,
particularly in the domains of astrophysics and cosmology. Additionally,
gaining a comprehensive understanding of the thermodynamic behavior of
relativistic particles, with due consideration of gravitational effects
(key-10, ; key-11, ), presents a promising avenue for exploration. Moreover,
studying related characteristics, such as fundamental statistical variables,
further enhances the potential to draw important and necessary conclusions
within the context of unraveling the enigmatic quantum behavior of gravity.

Topological defects (domain walls, cosmic strings, monopoles, and textures)
have been a focus of intense research over the last few decades, and they
continue to be one of the most active disciplines in condensed matter physics,
cosmology, astrophysics, and elementary particle models. These structures are
thought to have originated due to the Kibble mechanism (key-12, ; key-13, ;
key-14, ), in which defects occur in symmetry-breaking phase transitions
during the early universe’s cooling (key-15, ; key-16, ). The particular flaw
in question is cosmic strings ( for more reviews, we refer the reader to
(key-17, )). These items (whether static or revolving) can have discernible
impacts. These, for example, offer a viable method for seeding galaxy
formation and the 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. Moreover, 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 essential
instrument in many fields of theoretical physics (key-18, ). It is a well-
studied, absolutely solvable model that may be used to examine numerous
complicated issues within the context of quantum mechanics (key-19, ).
addition relativistic extension of the quantum harmonic oscillator gives a
useful model for understanding a wide range of molecular, atomic, and nuclear
interactions. Moreover, the property of having a complete set of exact
analytical solutions when working with such a model can give birth to
radically different interpretations of many mathematical and physical events,
and hence related applications can be realized via the underlying formulation.

The behavior of various relativistic quantum systems is now acknowledged to be
critically dependent on the so-called Dirac oscillator (DO) as Itô et al.
(key-20, ) specified in previous developments of spin-1/2 particle dynamics
with a linear trajectory. They demonstrated that the non-relativistic limit of
this system leads to an ordinary harmonic oscillator with a substantial spin-
orbit coupling term. Actually, Moshinsky and Szczepaniak (key-21, )
demonstrated that the DO might be derived from the free Dirac equation by
introducing an external linear potential through a minimum replacement of the
momentum operator $\hat{p}_{\mu}\rightarrow\hat{p}_{\mu}-iM\omega\beta
x_{\mu}$, with $\mu=0,1,2,3,\ldots$. It is worth noting that, in addition to
the theoretical focus on researching the DO, useful insights may be achieved
by examining physical interpretation, which is undoubtedly necessary for
comprehending many relevant applications Silva1 ; Silva2 ; Silva3 ; Silva4 ;
Silva5 ; boumali1 ; boumali2 .

Inspired by the DO, a similar formalism for the case of bosonic particles was
presented, and it was dubbed a Klein-Gordon oscillator (KGO) (key-23, ;
key-24, ). Several researchers have lately been working on the covariant
version of this model in curved spacetimes and in various configurations.
There are various contributions to the issue of relativistic quantum movements
of scalar and vector particles under gravitational influences caused by
different curved space-time geometries. For example, the problem of the
interaction between KGO coupled harmonically with topological defects in
Kaluza-Klein theory is examined in Ref. (key-24, ). Ref. (key-25, )
investigates the relativistic quantum dynamics of spin-0 particles in a
spinning cosmic string space-time with Coulomb-type scalar and vector
potentials. Additionally, 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-26, ). The authors of Ref.
(key-27, ) recently examined the KGO in cosmic string space-time and
investigated the effects of the rotating frame and non-commutativity in
momentum space. Additionally, Ref. (key-28, ) investigated the Relativistic
scalar charged particle in a rotating cosmic string space-time with Cornell-
type potential and Aharonov-Bohm effect. Efforts have been made by several
writers (key-30, ; key-31, ; key-32, ; boumali3, ; boumali4, ; boumali5, ) to
explore the relativistic spin-0, spin-1 bosons, and spin-1/2 fermions wave
functions and their time evolution using the Hamiltonian form, i.e.,
possessing Schrodinger-type equations. In this regard, the so-called FV
equations (key-33, ; key-34, ) are of special relevance. These equations were
originally designed to allow for a relativistic single-particle interpretation
of the second-order KG equation. In the latter scenario, FV equations result
from dividing the KG wave function into two components to produce an equation
with a first-order time derivative. Several investigations have been made in
recent decades with the goal of examining the relativistic dynamical features
of single particles and solving their wave equations using the FV technique
(e.g., Refs. (key-35, ; key-36, ; key-37, ; key-38, ; key-39, ; key-40, ;
key-41, ; key-42, ; key-43, ; key-44, ) and other related references cited
therein).

As mentioned in Ref. (key-44, ), the technique for introducing a scalar
potential into the KG equation is as follows. In In the same way, the
electromagnetic 4-vector potential is introduced. This is achieved by
modifying the momentum operator$p_{\alpha}=i\partial_{\alpha}$ to
$p_{i}\rightarrow p_{i}-qA_{i}\left(x\right)$. Another method was described in
Ref. (key-45, ), which included a modification in the mass of the particle as
$M\rightarrow M+S(\mathbf{r},t)$, where $S(\mathbf{r},t)$ indicates the scalar
potential. In recent decades, researchers have investigated the behavior of a
Dirac particle in the presence of a static scalar potential and a Coulomb
potential (key-45, ) as well as a relativistic scalar particle in cosmic
string spacetime (key-46, ). Bouzenada et al. (key-48, ) investigate the FVO
case in spinning cosmic string space-time and discuss some findings (thermal
properties and density of this system). This same model has also been studied
in the cosmic string space-time with a screw dislocation in the presence of
coulomb-type potential (key-51, ).

The structure of this paper is as follows. In Sec. II, we present a brief
review of the VF formalism emphasizing the relevant properties that are used
in our work. In. Sec. III, in the spinning cosmic string spacetime. In Sec.
IV, we apply the KG equation in the VF representation to three physical
models: (a) we study the case where the particle is confined to a hard-wall
potential, (b) motion in the presence of a Coulomb-type potential, and (c)
particle interacting with the KG oscillator. Our conclusions are presented in
Sec. V. Throughout the paper, we use natural units $\hbar=c=G=K_{B}=1$, and
our metric convention is $(+,-,-,-$).

##  II An overview of the Feshbach-Villars representation

This section focuses on the relativistic quantum depiction of a spin-0
particle moving through Minkowski space-time with the metric tensor
$\eta_{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right)$. We begin with the
conventional covariant KG equation to a scalar massive particle given by
(key-43, ; key-44, )

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

where $M$ is the mass of the particle, and the minimally-coupled covariant
derivative is represented as $D_{\mu}=i\left(p_{\mu}-eA_{\mu}\right)$. Here,
$p_{\mu}=\left(E,-p_{i}\right)$ denotes the canonical four-momentum,
$A_{\mu}=\left(A_{0},-A_{i}\right)$ represents the electromagnetic four-
potential, and $e$ represents the electric charge.

It is important to note that at this juncture, it can be emphasized that
Equation (1) has the capability to be restated in the form of a Hamiltonian
equation, involving the first derivative with respect to time, akin to a
Schrödinger-type equation

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

The Hamiltonian $\mathcal{H}$ can be achieved using the FV linearization
technique, which involves transforming Eq. (1) into a first-order differential
equation with respect to time. To achieve this, we introduce the two-component
wave function (key-43, ; key-44, )

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

where $\psi(x,t)$ satisfies the KG wave equation, and
$\mathcal{D}=\frac{\partial}{\partial t}+ieA_{0}(x)$. The transformation given
in Eq. (3) involves introducing wave functions that fulfill the specified
conditions

$\psi=\phi_{1}+\phi_{2},\qquad
i\mathcal{D}\psi=M\left(\phi_{1}-\phi_{2}\right).$ (4)

To facilitate our subsequent examination, it is advantageous to express
$\phi_{1}$ and $\phi_{2}$ as

$\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{2M}\left[M+i\frac{\partial}{\partial
t}-eA_{0}\right]\psi,$ (5) $\displaystyle\phi_{2}$
$\displaystyle=\frac{1}{2M}\left[M-i\frac{\partial}{\partial
t}+eA_{0}\right]\psi.$ (6)

Substituting the equations of both $\phi_{1}$ and $\phi_{2}$ in (3) and (1) ,
we get

$\displaystyle\left[i\frac{\partial}{\partial
t}-eA_{0}\right]\left(\phi_{1}+\phi_{2}\right)$
$\displaystyle=M\left(\phi_{1}-\phi_{2}\right),$ (7)
$\displaystyle\left[i\frac{\partial}{\partial
t}-eA_{0}\right]\left(\phi_{1}-\phi_{2}\right)$
$\displaystyle=\left[\frac{\left(p_{i}-eA_{i}\right)^{2}}{M}+M\right]\left(\phi_{1}+\phi_{2}\right).$
(8)

By adding and subtracting these two equations, we arrive at a system of
interconnected first-order differential equations with respect to time

$\displaystyle\frac{\left(p_{i}-eA_{i}\right)^{2}}{2M}\left(\phi_{1}+\phi_{2}\right)+\left(M+eA_{0}\right)\phi_{1}$
$\displaystyle=i\frac{\partial\phi_{1}}{\partial t},$ (9)
$\displaystyle-\frac{\left(p_{i}-eA_{i}\right)^{2}}{2M}\left(\phi_{1}+\phi_{2}\right)-\left(M-eA_{0}\right)\phi_{2}$
$\displaystyle=i\frac{\partial\phi_{2}}{\partial t}.$ (10)

By employing Eqs. (9) and (10), the Hamiltonian for a scalar particle under
the influence of electromagnetic interaction in the FV formalism can be
expressed as

$\mathcal{H}_{FV}=\left(\tau_{z}+i\tau_{y}\right)\frac{\left(p_{i}-eA_{i}\right)^{2}}{2M}+M\tau_{z}+eA_{0}(x).$
(11)

The Pauli matrices in the conventional $2\times 2$ form, denoted by
$\tau_{i},\left(i=x,y,z\right)$, are defined as

$\tau_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\tau_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\tau_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (12)

It is important to highlight that the Hamiltonian (11) satisfies the condition
of generalized Hermiticity. In other words, the Hamiltonian $\mathcal{H}$ is
considered pseudo-Hermitian if there exists an invertible, Hermitian, linear
operator $\beta$ such that the following condition is met:
$\mathcal{H}^{\dagger}=\beta\mathcal{H}\beta^{-1}$ (key-44, ). Thus, we write

$\mathcal{H}_{FV}=\tau_{z}\mathcal{H}_{FV}^{\dagger}\tau_{z},\qquad\mathcal{H}_{FV}^{\dagger}=\tau_{z}\mathcal{H}_{FV}\tau_{z}.$
(13)

In the case of free particle propagation, where no interaction is considered
(i.e., $A_{\mu}=0$), the one-dimensional FV Hamiltonian can be simplified to
the following form:

$\mathcal{H}_{0}=\left(\tau_{z}+i\tau_{y}\right)\frac{p_{x}^{2}}{2M}+M\tau_{z}.$
(14)

The solutions to the free Hamiltonian, which do not depend on time, correspond
to stationary states. Assuming a solution in the form mentioned in Ref.
(key-35, ), we write

$\Phi\left(x,t\right)=\Phi\left(x\right)e^{-iEt}=\left(\begin{array}[]{c}\phi_{1}\left(x\right)\\\
\phi_{2}\left(x\right)\end{array}\right)e^{-iEt},$ (15)

with $E$ being the energy of the particle. Thus, Eq. (2) can then be written
as

$\mathcal{H}_{0}\Phi\left(x\right)=E\Phi\left(x\right).$ (16)

The one-dimensional FV equation for a free relativistic spin-0 particle is an
alternative Schrödinger-type equation to the KG equation. Its purpose is to
provide a different perspective on the dynamics of the particle. In the
subsequent analysis, this approach will be employed to derive solutions for
wave equations in curved space-time, specifically focusing on the Non-inertial
cosmic string. By utilizing this framework, a deeper understanding of the
behavior of particles in the presence of curved space-time can be achieved.

##  III Quantum Dynamics of Spin-0 Particle in Non-inertial Cosmic String
Space-time: The FV Representation

The equations governing the behavior of a scalar particle in a Riemannian
spacetime, defined by the metric tensor $g_{\mu\nu}$, can be obtained by
reformulating the Klein-Gordon equation. This approach, extensively discussed
in various textbooks (key-9, ; key-10, ; key-40, ; key-41, ; key-48, ; key-49,
; key-52, 52), allows for a deeper understanding of the dynamics of particles
in curved spacetime. The KG equation is given by

$\left(\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right)+M^{2}-\xi
R\right)\Phi(x,t)=0.$ (17)

Here, the Laplace-Beltrami operator
$\square=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right)$
plays a crucial role. It represents the differential operator used in the KG
equation in the curved space-time of the cosmic string, whose line element is
defined later. Additionally, $\xi$ denotes a real dimensionless coupling
constant, and $R$ corresponds to the Ricci scalar curvature defined by
$R=g^{\mu\nu}R_{\mu\nu}$, with $R_{\mu\nu}$ representing the Ricci curvature
tensor. The inverse metric tensor is denoted by $g^{\mu\nu}$, and
$g=\det\left(g_{\mu\nu}\right)$ is the determinant of the metric tensor. These
quantities collectively contribute to formulating the KG equation in the
cosmic string space-time.

In the pursuit of understanding the quantum dynamics of spin-0 particles in
space-time influenced by the non-inertial effects of a cosmic string, our
current objective is to determine the equation of motion governing their
motion. To accomplish this, we will employ the FV representation. However,
before delving into specific physical models, we first derive the KG wave
equation for a free relativistic scalar particle propagating in a cosmic
string space-time that is both static and cylindrically symmetric. This serves
as the foundation for further investigations into the behavior of particles in
the presence of cosmic strings.

The line element that defines the metric of a cosmic string in a
(3+1)-dimensional spacetime can be expressed in a general form as (key-40, ;
key-41, ; key-48, ; key-49, ; key-52, 52)

$\displaystyle ds^{2}$ $\displaystyle=g_{\mu\nu}dx^{\mu}dx^{\nu},$
$\displaystyle=dt^{2}-dr^{2}-\alpha^{2}r^{2}d\varphi^{2}-dz^{2},$ (18)

which is an exact solution to Einstein’s field equations for $0\leq\mu<1/4$.
Furthermore, by setting $\varphi^{\prime}=\alpha\varphi$, it represents a flat
conical exterior space with an angle deficit $\delta\phi=8\pi\mu$). In Eq.
(18), $-\infty\leq t\leq+\infty$, $r\geq 0$, $0\leq\varphi\leq 2\pi$,
$-\infty\leq z\leq+\infty,$ and $\alpha\in[0,1[$ is the parameter which
determines the angular deficit $\delta\varphi=2\pi(1-\alpha)$, and it is
related to the linear mass density $\mu$ of the string by $\alpha=1-4\mu$. The
line element (18) provides a mathematical representation of the spacetime
geometry associated with the cosmic string, capturing the relationships
between distances and intervals within this particular curved space-time. By
considering the specific structure of this line element, one can examine the
properties and characteristics of the cosmic string metric in a
(3+1)-dimensional setting.

Alternatively, we can consider a string with a linear mass density equal to
along the $z$ axis, with the Lorentz metric $ds^{2}=dt^{2}-dx^{\prime
2}-dy^{\prime 2}-dz^{\prime 2}$ and the coordinate changes
$x^{\prime}=R\cos(\alpha\bm{\Phi}),y=R\sin(\alpha\bm{\Phi}),z=\bm{Z}$, and
$t=\bm{T}$, which leads to the line element of a cosmic string space-time with
cylindrical coordinates (key-m53, 53, 54, 55, 56, 57, 58, 59)

$ds^{2}=d\bm{T}^{2}-d\bm{R}^{2}-\left(\alpha\bm{R}\right)^{2}d\bm{\Phi}^{2}-d\bm{Z}^{2}.$
(19)

In addition to the presence of non-inertial effects, we analyze a frame that
rotates evenly with a constant angular velocity $\Omega$. The line element
that describes this system is written as (key-m53, 53, 54, 55, 56, 57, 58, 59)

$ds^{2}=\left(1-\alpha^{2}\Omega^{2}r^{2}\right)dt^{2}-2\Omega\alpha^{2}r^{2}d\varphi
dt-dr^{2}-\alpha^{2}r^{2}d\varphi^{2}-dz^{2}.$ (20)

Since we do not want the term $g^{00}$ to become positive, we shall require
that

$0<r<\frac{1}{\alpha\Omega}.$ (21)

This ensures that the line element’s radial coordinate (20) is defined inside
the range. The metric and inverse metric tensor components are given,
respectively, by (key-m53, 53, 54, 55, 56, 57, 58, 59)

$g_{\mu\nu}=\left(\begin{array}[]{cccc}\left(1-\left(\alpha\Omega
r\right)^{2}\right)&0&-\Omega\left(\alpha r\right)^{2}&0\\\ 0&-1&0&0\\\
-\Omega\left(\alpha r\right)^{2}&0&-\left(\alpha r\right)^{2}&0\\\
0&0&0&-1\end{array}\right),\quad
g^{\mu\nu}=\left(\begin{array}[]{cccc}1&0&-\Omega&0\\\ 0&-1&0&0\\\
-\Omega&0&\left(\Omega^{2}-\frac{1}{\left(\alpha r\right)^{2}}\right)&0\\\
0&0&0&-1\end{array}\right).$ (22)

It is worth noting that the problem of spinless heavy particles in the
geometry formed by a non-inertial cosmic string background has been studied in
various studies (see, for example, (key-m53, 53, 54, 55, 56, 57, 58, 59)). To
obtain the FV form of the KG wave equation in curved manifolds, we will use
the approach described in references (key-44, ). We employ the generalized
Feshbach-Villars transformation (GFVT) to describe both massive and massless
particles (an equivalent transformation was presented earlier in Ref. (key-55,
)). The components of the wave function $\Phi$ in the GFVT are provided by
(key-44, )

$\psi=\phi_{1}+\phi_{2},\qquad
i\tilde{\mathcal{D}}\psi=\mathcal{N}\left(\phi_{1}-\phi_{2}\right),$ (23)

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

$\mathcal{Y}=\frac{1}{2g^{00}\sqrt{-g}}\left\\{\partial_{i},\sqrt{-g}g^{0i}\right\\}.$
(24)

The curly bracket in Eq. (24) denotes the anti-commutator. For the above-
mentioned transformation, the Hamiltonian reads

$\displaystyle\mathcal{H}_{GFVT}$
$\displaystyle=\tau_{z}\left(\frac{\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)+i\tau_{y}\left(\frac{-\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)-i\mathcal{Y},$
$\displaystyle\mathcal{T}$
$\displaystyle=\frac{1}{g^{00}\sqrt{-g}}\partial_{i}\sqrt{-g}g^{ij}\partial_{j}+\frac{M^{2}-\xi
R}{g^{00}}-\mathcal{Y}^{2},\;\text{with}\;(i,j=1,2,3).$ (25)

Note that for $\mathcal{N}=M$, the original FV transformations are satisfied.

Using the metric 20, we can verify that $R=0$, implying that space-time is
locally flat (there is no local gravity), and so the coupling term is
vanishing. (The condition $\xi=0$ is known as minimum coupling. Yet, in
massless theory, $\xi$ equals 1/6. (in 4 dimensions). The equations of motion
are then conformally invariant in this latter instance.). If $\mathcal{Y}\neq
0,$ then, we obtain

$\psi=\phi_{1}+\phi_{2},\qquad
i\mathcal{D}^{\prime}\psi=\mathcal{N}\left(\phi_{1}-\phi_{2}\right),$ (26)

with $\mathcal{D}^{\prime}=\frac{\partial}{\partial t}+\mathcal{Y^{\prime}}.$
The next step is to extend the method employed in Sec. II to the situation of
spinning cosmic strings. The GFVT is used to construct the equations of motion
for this issue, which are then solved to give the wave functions and energy
spectra. According to the Refs. (key-52, 52, ), identifying the Hamiltonian
leads to the concept of using a two-component formulation of the KG-type
fields. To recast Eq. (28) in the Hamiltonian form, additional definitions for
the quantities described in Eqs. (22) must be introduced Eqs. (24) to (32)
(key-54, ). In this way, we write

$\mathcal{H}_{GFVT}^{\prime}=\tau_{z}\left(\frac{\mathcal{N}^{2}+\mathcal{T^{\prime}}}{2\mathcal{N}}\right)+i\tau_{y}\left(\frac{-\mathcal{N}^{2}+\mathcal{T^{\prime}}}{2\mathcal{N}}\right)-i\mathcal{Y}^{\prime},$
(27)

with

$\displaystyle\mathcal{T}^{\prime}$
$\displaystyle=\partial_{i}\frac{G^{ij}}{g^{00}}\partial_{j}+\frac{M^{2}-\xi
R}{g^{00}}+\frac{1}{\mathcal{F}}\nabla_{i}\left(\sqrt{-g}G^{ij}\right)\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\sqrt{\frac{\sqrt{-g}}{g^{00}}}G^{ij}\nabla_{i}\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\frac{1}{4\mathcal{F}^{4}}\left[\nabla_{i}\left(\mathcal{U}^{i}\right)\right]^{2}$
$\displaystyle-\frac{1}{2\mathcal{F}^{2}}\nabla_{i}\left(\frac{g^{0i}}{g^{00}}\right)\nabla_{j}\left(\mathcal{U}^{i}\right)-\frac{g^{0i}}{2g^{00}\mathcal{F}^{2}}\nabla_{i}\nabla_{j}\left(\mathcal{U}^{i}\right),$
(28)

and

$\mathcal{Y}^{\prime}=\frac{1}{2}\left\\{\partial_{i},\sqrt{-g}\frac{g^{0i}}{g^{00}}\right\\},$
(29)

where

$G^{ij}=g^{ij}-\frac{g^{0i}g^{0j}}{g^{00}},\;\;\mathcal{F}=\sqrt{g^{00}\sqrt{-g}},\;\;\mathcal{U}^{i}=\sqrt{-g}g^{0i}.$
(30)

Reference (key-54, ) demonstrates that the transformations (27), (28), and
(29) with the definitions in Eq. (30) are precise and encompass all inertial
and gravitational fields. It should be emphasized that using these precise
transformations guarantees that the block-diagonal form of the Hamiltonian
$\mathcal{H}_{GFVT}$ is obtained, which is independent of the parameter
$\mathcal{N}$. The eigenvalues of the operator $\mathcal{Y}^{\prime}$ for the
geometry in Eq. (20) are given by

$\mathcal{Y}^{\prime}\Psi\left(t,r,\varphi\right)=\frac{1}{2}\left\\{\partial_{2},\sqrt{-g}\frac{g^{02}}{g^{00}}\right\\}\Psi\left(t,r,\varphi\right)=\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\Psi\left(t,r,\varphi\right),$
(31)

where the field $\Psi$ obeys the non-unitary transformation
$\Psi\equiv\Phi^{\prime}=\mathcal{F}\Phi$, which permits to obtain pseudo-
Hermitian Hamiltonian
$\mathcal{H}_{GFVT}^{\prime}=\mathcal{F}\mathcal{H}_{GFVT}^{\prime}\mathcal{F}^{-1}$,
with
$\mathcal{H}_{GFVT}^{\prime}=\tau_{z}\left(\mathcal{H}_{GFVT}^{\prime}\right)^{\dagger}\tau_{z}.$
After some mathematical calculations, Eq. (28) takes the form

$\mathcal{T}^{\prime}=-\frac{1}{\mathcal{F}}\left[\left(\frac{\partial}{\partial
r}\right)\left(\sqrt{-g}\frac{\partial}{\partial
r}\right)+\sqrt{-g}\left(\varOmega^{2}-\frac{1}{\alpha^{2}r^{2}}\right)\frac{\partial^{2}}{\partial\varphi^{2}}+\sqrt{-g}\frac{\partial^{2}}{\partial
z^{2}}\right]\frac{1}{\mathcal{F}}+\frac{M^{2}}{g^{00}}-\left(\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\right)^{2},$
(32)

where for the metric (20) the Ricci scalar vanishes i.e, $R=0$. The goal of
using the Hamiltonian (27) and then using the Hamiltonian (30) is to find
solutions in the form of the Schrödinger equation, such as

$\mathcal{H}_{GFVT}^{\prime}\Psi^{\prime}(\bm{x})=i\,\frac{d}{dt}\Psi^{\prime}(\bm{x}),\qquad\bm{x}\equiv(t,r,\varphi,z).$
(33)

Consider the ansatz below for the wave function to solve this eigenvalue
problem

$\Psi(\bm{x})=\mathcal{F}\Phi\left(\bm{x}\right)=\mathcal{F}\left(\begin{array}[]{c}\phi_{1}\left(\bm{r}\right)\\\
\phi_{2}\left(\bm{r}\right)\end{array}\right)e^{-\left(iEt-j\varphi-
k_{z}z\right)},$ (34)

where $j=0,\pm 1,\pm 2,\pm 3,\ldots$ is the angular momentum quantum number,
and $k_{z}\in\left[-\infty,\infty\right]$. By replacing Eq. (34) into Eq.
(33), we obtain the differential equations

$\displaystyle\mathcal{N}^{2}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+2\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\phi_{1}$
$\displaystyle=2\mathcal{N}E\mathcal{F}\phi_{1},$ (35)
$\displaystyle\mathcal{N}^{2}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)-\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}+\phi_{2}\right)+2\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\phi_{2}$
$\displaystyle=2\mathcal{N}E\mathcal{F}\phi_{2}.$

Adding and subtracting these equations leads, after simplification, to sets of
coupled equations for $\phi_{1}$ and $\phi_{2}$

$\displaystyle\mathcal{N}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\left(\phi_{1}+\phi_{2}\right)=E\mathcal{F}\left(\phi_{1}+\phi_{2}\right),$
(36) $\displaystyle-$
$\displaystyle\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}+\phi_{2}\right)+\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\left(\phi_{1}-\phi_{2}\right)=\mathcal{N}E\mathcal{F}\left(\phi_{1}-\phi_{2}\right),$
(37)

where we have used the result

$\phi_{1}=\frac{1}{\mathcal{N}}\left(E-\frac{g^{02}}{g^{00}}j\right)\phi_{2}=\frac{1}{\mathcal{N}}\left(E+\varOmega
j\right)\phi_{2}.$ (38)

After some simple algebraic manipulations, we arrive at the following second-
order differential equation for the radial function $\psi(r)$:

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\varsigma^{2}}{r^{2}}+\gamma^{2}\right]\varphi_{1}\left(r\right)=0,$
(39)

where we have defined the parameters $\varsigma^{2}=j^{2}/\alpha^{2}$ and
$\gamma^{2}=\left(E+\varOmega j\right)^{2}-M^{2}-k_{z}^{2}$. It can be shown
that Eq. (39) is a Bessel-type differential equation, and its general solution
is given by

$\psi\left(r\right)=A^{\prime}\,J_{\varsigma}\left(\gamma
r\right)+B^{\prime}N_{\varsigma}\left(\gamma r\right),$ (40)

where $J_{\varsigma}\left(\gamma r\right)$ and $N_{\varsigma}\left(\gamma
r\right)$ are the Bessel function of the first and second kind, and
$A^{\prime}$ and $B^{\prime}$ are arbitrary constants. In the next sections,
we make some applications of the above formalism. We shall consider models
already consolidated in the literature to make clear the physical implications
due to the non-inertial effects and the curvature parameter of the cosmic
string.

##  IV Applications

###  IV.1 Particle confined to a hard-wall potential

Studying the relativistic motion of a spin-0 particle confined to a hard-wall
potential is important in theoretical physics and provides us with very
interesting results. By exploring this scenario, we gain valuable insights
into the behavior of particles under extreme conditions and the interplay
between relativity and quantum mechanics. This study lets us deepen our
understanding of fundamental principles, refine theoretical frameworks, and
develop computational techniques. Moreover, it serves as a stepping stone for
comprehending more complex systems, guiding us toward a more complete
understanding of the universe. Through this investigation, we unveil the
intricate dynamics of spin-0 particles and contribute to the broader
advancement of physics as a whole. For the simple model described by the
solution of Eq. (39), we must consider the physically acceptable solution in
the region of interest. In this case, since solution (40) must be regular at
the origin and function $N_{\varsigma}\left(\gamma r\right)$ is not well
behaved in this region, we must make $B=0$ in the solution (40). The relevant
solution is

$\psi\left(r\right)=A^{\prime}\,J_{\varsigma}\left(\gamma r\right).$ (41)

Figure 1: Sketch of the energies (Eq. (45)) as a function of $\Omega$ for
different values of $n$ and $\alpha$. In (a), the profile for $n=0$ and $j=0$,
(b) $n=1$ and $j=0$, (c) $n=1$ and $j=1$, and (d) $n=1$ and $j=-1$. We use
$M=1$, $k_{z}=1$, and $\kappa=2.5$. A manifestation due to curvature effects
is observed when $\alpha$ is decreased, which implies $|E_{n,j}^{(\pm})|$
increasing.

To have a normalized wave function in the region established by the range
(21), we must require that

$\psi\left(r\rightarrow r_{0}=\frac{1}{\alpha\Omega}\right)=0.$ (42)

This condition remains valid for $r_{0}\gg 1$ and fixed values of $\varsigma$
and $\gamma$. In this limit, we can write the solution (41) in the form

$\psi\left(\bm{r}\right)=|\mathcal{C}_{1}|\sqrt{\frac{2}{\pi\gamma
r_{0}}}\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\varphi-
k_{z}z\right)}\,\cos\left(\gamma
r_{0}-\frac{\varsigma\pi}{2}-\frac{\pi}{4}\right).$ (43)

By applying condition (58) to the solution (43), we obtain the relation

$\gamma
r_{0}-\frac{\varsigma\pi}{2}-\frac{\pi}{4}=\frac{n\pi}{2},\text{with}\,n\in\mathbb{Z^{*}},$
(44)

which solved for $E$ provides

$E_{n,j}^{(\pm)}=-j\Omega\pm\sqrt{\left(\frac{1}{2}\pi\Omega\left|j\right|+\frac{1}{4}\pi\Omega\alpha+\frac{1}{2}n\pi\Omega\alpha\right)^{2}+k_{z}^{2}+M^{2}},$
(45)

which represents the particle’s energy. By studying these energies, we can
report some inherent characteristics of the model. We evaluate some particular
states of the energy (45) as a function of $\Omega$. All graphical analyses in
our article we use $M=1$, $k_{z}=1$, and $\kappa=2.5$ (Fig. 1). We can see
that $E_{n,j}$ exhibits different profiles when $\alpha$ changes, for example,
when $\alpha=0.2,0.4,0.6,0.8,1.0$, $|E_{0,1}^{(\pm)}|$ is symmetric about
$\Omega=0$, and increases when $\Omega$ is increased. When this occurs, it is
observed that the spectrum becomes asymmetric (Fig. 1(a)). These features are
also observed when we investigate $E_{1,0}$, but with the difference that
$|E_{0,1}^{(\pm)}|<|E_{1,0}^{(\pm)}|$ (Fig. 1(b)). An interesting feature is
observed when we study the energies for $n=1,j=1$ (Fig. 1(c)) and $n=1,j=-1$
(Fig. 1(d)). We observe that $|E_{1,1}^{(\pm)}|$ and $|E_{1,-1}^{(\pm)}|$
increase with $\Omega$. However, $|E_{1,1}^{(+)}|<|E_{1,-1}^{(+)}|$ while
$|E_{1,1}^{(-)}|>|E_{1,-1}^{(-)}|$. This manifestation is due to the combined
effects of rotation and the parameter $\alpha$.

###  IV.2 The Coulomb-type potential

In this section, we solve the KG equation with a Coulomb-like potential in the
FV formalism. Including this potential in the KG equation allows investigating
of long-range electromagnetic interactions and describing the interaction
between charged particles. This investigation is crucial for understanding
various physical phenomena, from atomic structure to elementary particle
physics. We must consider an attractive Coulomb-type potential to solve the
equation then and analyze fundamental properties such as bound states. We
include the potential in the equation of motion by substituting $E\rightarrow
E-V(r)$ into Eq. (39). The potential $V(r)$ is specified by

$V\left(r\right)=-\frac{\kappa}{r},$ (46)

where $\kappa$ is a positive arbitrary parameter. The radial equation to be
solved is

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\zeta^{2}}{r^{2}}+\frac{2\lambda}{r}-\gamma^{2}\right]\varphi\left(r\right)=0,$
(47)

where
$\zeta^{2}=\varsigma^{2}+\kappa^{2},\gamma^{2}=M^{2}+k_{z}^{2}-\left(\mathcal{E}+\varOmega
j\right)^{2}$, and $\lambda=\kappa\left(\mathcal{E}+\varOmega j\right)$. The
radial equation (47) is of the confluent hypergeometric type, and a solution
satisfying the asymptotic limits for both $r\rightarrow 0$ and
$r\rightarrow\infty$ can be found. Furthermore, this equation appears in
several problems solved in the literature, so we do not find it necessary to
present a detailed solution. Thus, using the results above, it can be shown
that the eigenfunctions and energy eigenvalues of Eq. (47) in the FV
representation are respectively given by

$\psi\left(\bm{r}\right)=C_{n}\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\zeta-
k_{z}z\right)}\,(2\gamma)^{\zeta+\frac{1}{2}}e^{-\gamma
r}r^{\zeta}\,_{1}F_{1}\left(-n,2\zeta+1,2\gamma r\right),$ (48)
$\mathcal{E}^{(\pm)}_{n,j}=-j\Omega\pm\frac{(2\zeta+2n+1)^{2}\left(k_{z}^{2}+M^{2}\right)}{\sqrt{(2\zeta+2n+1)^{2}\left(4\kappa^{2}+(2\zeta+2n+1)^{2}\right)\left(k_{z}^{2}+M^{2}\right)}}.$
(49)

Figure 2: Sketch of the energies (Eq. (49)) as a function of $\Omega$ for
different values of $n$ and $\alpha$. In (a), the profile for $n=0$ and $j=1$,
(b) $n=3$ and $j=1$, (c) $n=6$ and $j=1$,and (d) $n=9$ and $j=1$. We use
$M=1$, $k_{z}=1$, and $\kappa=2.5$. The states with increasing $n$ reveal an
approximation between the energy curves. When $\alpha$ is decreased,
$|E_{n,j}^{(\pm})|$ increases.

Figure 3: Sketch of the energies (Eq. (49)) as a function of $\Omega$ for
different values of $n$ and $\alpha$. In (a), the profile for $n=0$ and
$j=-1$, (b) $n=0$ and $j=-3$, (c) $n=0$ and $j=-5$; and (d) $n=0$ and $j=-7$.
We use $M=1$, $k_{z}=1$, and $\kappa=2.5$. In addition to the features present
in Fig. 2, when we set $n$ and consider different negative values of $m$, the
energies are defined in a rotation range that tends to decrease with negative
$m$.

Possible choices for the parameters in (49) can display different profiles.
For example, analyzing the states with $j=1$ and $n=0,3,6,9$, the positive and
negative values of $\mathcal{E}^{(\pm)}_{n,j}$ are shown in Fig. 2. The
curvature effects on the energies directly impact the spacing between the
energy curves, making this spacing somewhat smaller for increasingly larger
values of $n$ (Figs. 2(a)-(d)). The inset in Fig. 2(d) shows that such
spacings are maintained, and the energy has an approximately linear change
over small rotation intervals. For this particular configuration,
$|\mathcal{E}^{(\pm)}_{n,j}|$ decreases when $\Omega$ increases, and it exists
only in the interval $0<\Omega<1$. A second configuration is displayed in Fig.
3 for $n=0$ and $j=-1,-2,-3,-4$. For this particular configuration,
$|\mathcal{E}_{nj}^{\pm}|$ increases when $\alpha$ is reduced. However, when
$m$ takes on larger values, the range of $\Omega$ that defines the allowed
energies is reduced. This effect becomes clear when we look at the energy
profiles in Figs. 3(a)-3(d).

###  IV.3 The Feshbach-Villars Oscillator (FVO)

The final application of our study involves including the FVO in the equation
of motion. The FVO is a well-known, exactly solvable model in relativistic
quantum mechanics that have been constructed in both flat and curved spacetime
(key-48, ). This model combines the principles of special relativity with
quantum mechanics, providing insights into the dynamics of spin-0 particles in
a harmonic potential. By exploring the FVO, we gain a deeper understanding of
wave equations, energy spectra, and the quantization of fields. Moreover, this
study opens doors to investigating phenomena like particle creation and
annihilation, vacuum fluctuations, and the manifestation of symmetries. As an
extension, it is important to study the FVO in other physical contexts because
it helps us to expand our knowledge of fundamental physics, improve our
analytical capabilities to solve new equations of motion, and prepare the way
for future advances in quantum field theory. The FVO is introduced into
equation (28) and (29) by substituting
$\partial_{r}\rightarrow\partial_{r}+M\omega r$. So, we write

$\displaystyle\mathcal{T}^{\prime}$
$\displaystyle=-\frac{1}{\mathcal{F}}\left[\left(\frac{\partial}{\partial
r}+M\omega r\right)\left(\sqrt{-g}\left(\frac{\partial}{\partial r}-M\omega
r\right)\right)+\sqrt{-g}\left(\varOmega^{2}-\frac{1}{\alpha^{2}r^{2}}\right)\frac{\partial^{2}}{\partial\varphi^{2}}+\sqrt{-g}\frac{\partial^{2}}{\partial
z^{2}}\right]\frac{1}{\mathcal{F}}$
$\displaystyle+\frac{M^{2}}{g^{00}}-\left(\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\right)^{2}.$
(50) $\mathcal{Y}^{\prime}=0.$ (51)

Equation (50) can be solved by following the same steps as before. We shall
insert Eqs. (50) and (31) into the Hamiltonian (27). Subsequently, we assume
the solution (34) gives two coupled differential equations similar to those of
Eq. (35), but with different values of $\mathcal{T}^{\prime}$. We obtain the
radial equation

$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-M^{2}\omega^{2}r^{2}-\frac{\vartheta^{2}}{r^{2}}+\delta\right]\psi\left(r\right)=0,$
(52)

where we have defined the parameters

$\vartheta^{2}=\frac{j^{2}}{\alpha^{2}},\;\;\delta=\left(E+\varOmega
j\right)^{2}-M^{2}-k_{z}^{2}+M\omega.$ (53)

To solve Eq. (52), we introduce the new dimensionless variable
$\mathcal{U}=M\omega r^{2}$, and replace it in Eq. 52. The resulting equation
reads

$\left[\frac{d^{2}}{d\mathcal{U}^{2}}+\frac{1}{\mathcal{U}}\frac{d}{d\mathcal{U}}-\frac{\vartheta^{2}}{4\mathcal{U}^{2}}-\frac{1}{4}+\frac{\delta}{4M\omega\mathcal{U}}\right]\psi\left(\mathcal{U}\right)=0.$
(54)

Now, we introduce the new function

$\psi(\mathcal{U})=\mathcal{U}^{-\frac{1}{2}}\mathcal{J}\left(\mathcal{U}\right).$
(55)

With this substitution, Eq. (54) becomes

$\frac{d^{2}\mathcal{J}\left(\mathcal{U}\right)}{d\mathcal{\mathcal{U}}^{2}}+\left[-\frac{1}{4}+\frac{\delta}{4M\omega\mathcal{\mathcal{U}}}+\frac{\frac{1}{4}-\left(\frac{\vartheta}{2}\right)^{2}}{\mathcal{\mathcal{U}}^{2}}\right]\mathcal{J}\left(\mathcal{U}\right)=0.$
(56)

It can be shown that Eq. (56) is of the confluent hypergeometric type, whose
solution is given in terms of Kummer functions. Thus, using the definition
(34), the solution of Eq. (56) can be written as

$\psi\left(\bm{r}\right)=|\mathcal{C}_{3}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)\left(M\omega
r^{2}\right)^{\frac{\left|\vartheta\right|}{2}}e^{-\frac{M\omega}{2}r^{2}}e^{-i\left(Et-j\varphi-
ik_{z}z\right)}\,_{1}F_{1}\left(\frac{|\vartheta|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},|\vartheta|+1,M\omega
r^{2}\right),$ (57)

where the parameters $\vartheta$ and $\delta$ are defined in Eq. (53). The
energies of the KGO are found by the relation

$\frac{|\vartheta|}{2}-\frac{\delta}{4M\omega}+\frac{1}{2}=-n.$ (58)

Figure 4: Sketch of the energies (Eq. (59)) as a function of $\Omega$ for
different values of $n$ and $\alpha$. We use $M=1$, $k_{z}=1$, and $\omega=1$.

Figure 5: Sketch of the energies (Eq. (59)) as a function of $\omega$ for
different values of $\alpha$. In (a), we plot for $n=0,j=1$ and b $n=0,j=-1$.
We use $M=1$, $k_{z}=1$, and $\Omega=1$. Figure 6: Sketch of the energies (Eq.
(59)) as a function of $\omega$ for $n=0,j=-1,\Omega=-1$ and different values
of $\alpha$. We use $M=1$, $k_{z}=1$.

Substituting back the parameters $\vartheta$ and $\delta$ in Eq. (58) and
solving the resulting equation for $E$, we obtain the energy levels

$E^{\pm}(n)=-\Omega
j\pm\sqrt{2M\omega\left(2n+\frac{\left|j\right|}{\alpha}+1\right)+M^{2}+k_{z}^{2}}\,,$
(59)

and the corresponding wave functions are

$\psi\left(\bm{r}\right)=|\mathcal{C}_{3}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\\
1-\frac{E}{\mathcal{N}}\end{array}\right)\left(M\omega
r^{2}\right)^{\frac{\left|\vartheta\right|}{2}}e^{-\frac{M\omega}{2}r^{2}}e^{-i\left(Et-j\varphi-
ik_{z}z\right)}\,_{1}F_{1}\left(-n,|\vartheta|+1,M\omega r^{2}\right).$ (60)

It is immediate to see that the plot of (59) as a function of $\Omega$ is
linear. For the particular state $\mathcal{E}_{0,1}^{(\pm)}$ with $\omega=1$,
we find that $\left|\mathcal{E}_{0,1}^{(\pm)}\right|$ increases when $\alpha$
decreases. On the other hand, when we analyze the profile of (59) as a
function of the oscillator frequency $\omega$, we find that
$\left|\mathcal{E}_{0,1}^{(\pm)}\right|$ e
$\left|\mathcal{E}_{0,-1}^{(\pm)}\right|$ increase with $\alpha$ decreasing
(Fig. 5). Furthermore,
$\left|\mathcal{E}_{0,1}^{(+)}\right|<\left|\mathcal{E}_{0,-1}^{(+)}\right|$
while $\left|\mathcal{E}_{0,1}^{(-)}\right|$ $>$
$\left|\mathcal{E}_{0,-1}^{(+)}\right|$ (see Figs. 5(a)-(b) for comparison).
When we assume negative values of $\Omega$ in Fig. 5(b), for example,
$\Omega=-1$, the oscillator energies for $\alpha=0.2$ are or all negative (see
the red solid line in Fig. 6). The energies of the FV oscillator are affected
by the curvature, specifically by the term $|j|/\alpha$.

##  V Conclusion

We have investigated the relativistic quantum motion of a spinless particle by
solving the KG equation in the FV representation. We have considered the
particle under spin effects in the space-time cosmic string. We briefly
reviewed the VF formalism and subsequently used the results to find the KG
equation in the space-time of the spinning cosmic string. To illustrate the
importance of the model studied, we have performed three applications. First,
we study the KG equation for the particular case where the particle is
confined to a hard-wall potential, next we study the motion of the particle in
the presence of a Coulomb-type potential, and we finish by studying the KG
equation with the inclusion of the KGO. Wave functions and energies for the
three models were determined. We have shown through graphical sketches of the
energies that for certain values of rotation and curvature (through the
$\alpha$ parameter), the particle’s energy shows limited profiles, especially
with the presence of rotation. We have argued that curvature strongly impacts
the magnitude of the energy levels. With these results, we would like to point
out that other physical properties can be studied, such as optical and
thermodynamic properties and possible connections to other physical systems in
condensed matter.

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