###### Abstract

We investigate the quantum dynamics of a harmonic oscillator (HO) system
within the framework of topologically charged Eddington-inspired Born-Infeld
(EiBI) gravity space-time. Additionally, we incorporate the Wu-Yang magnetic
monopole (WYMM) into the quantum system and analyze the influences of modified
gravity, the topological charge, and WYMM on this HO quantum mechanical
system. Using analytical methods, we derive the energy eigenvalues and the
corresponding eigenfunctions of the HO. Moreover, we consider a scenario where
the EiBI-gravity parameter is absent and introduce an inverse square
potential, including the WYMM, into the HO system. Our findings show
significant deviations in the behavior of the HO system compared to the
traditional quantum mechanical HO model, including alterations in the bound-
state spectra and eigenfunctions. These results provide valuable insights into
the interplay between quantum mechanical problems and alternative
gravitational theories.

Harmonic oscillator system in topologically charged Eddington-inspired Born-
Infeld gravity space-time and Wu-Yang magnetic monopole

Faizuddin<EMAIL_ADDRESS>
Department of Physics, University of Science & Technology Meghalaya, Ri-Bhoi,
Meghalaya, 793101, India
Abdelmalek<EMAIL_ADDRESS>
Laboratory of Theoretical and Applied Physics, Echahid Cheikh Larbi Tebessi
University, Algeria

Keywords: Modified gravity theories; Non-relativistic wave equations: Harmonic
Oscillator; solutions of wave equations; bound-states; magnetic monopoles;
special functions

PACS: 03.30.+p; 03.50.Kk; 04.62.+v; 11.27.+d; 03.65.Pm; 03.65.Ge; 02.30.Gp

##  1 Introduction

When Albert Einstein developed General Relativity in 1915 [1], it
revolutionized our understanding of gravity by characterizing it as the
curvature of space-time caused by mass and energy, rather than as a force.
This theory provides a comprehensive framework for understanding the large-
scale structure of the universe, extending the principles of Special
Relativity to include the effects of gravity [2]. According to General
Relativity, massive objects such as planets and stars warp the space-time
around them. This curvature affects the motion of other objects, producing the
effect we perceive as gravitational attraction. General Relativity has
profound implications for our understanding of black holes, the expansion of
the universe, and the nature of time and space. It remains a cornerstone of
modern physics.

Two of the most fundamental theories in modern physics are General Relativity
[3] and Quantum Mechanics [4], each describing unique aspects of the natural
world. Quantum Mechanics is founded on the principles of wave-particle duality
[5], the quantization of energy [6], and probabilistic outcomes [7], governing
the behavior of particles at the smallest scales, including atoms and
subatomic particles [8]. In contrast, General Relativity describes the
macroscopic world by using the curvature of space-time to explain
gravitational interactions between massive objects. Despite their successes,
these two frameworks are fundamentally irreconcilable; General Relativity
operates on a deterministic and continuous space-time fabric, while Quantum
Mechanics is inherently discrete and probabilistic. The quest to unify these
theories into a single coherent framework, often referred to as quantum
gravity, remains one of the most profound challenges in modern physics.
Advances in this area promise to deepen our understanding of phenomena such as
black holes [9, 10], the early universe, and the fundamental nature of space
and time itself. Various approaches [11], including string theory [12], loop
quantum gravity [13], and modifications like Born-Infeld gravity [14, 15, 16,
17], aim to bridge this gap, suggesting that the universe at its most
fundamental level might be governed by principles that reconcile these two
seemingly disparate theories.

An intriguing and intricate field of theoretical physics that combines ideas
from General Relativity and Quantum Mechanics is the study of the harmonic
oscillator in curved space-time [18, 19]. Unlike the behavior of a basic
harmonic oscillator in flat space-time [20], the curvature of space-time
itself significantly influences the behavior of such a system in a curved
background [21]. This curvature can be caused by the presence of massive
objects or more complex gravitational fields [22, 23]. The wave function and
energy spectrum of the oscillator field are altered by this curvature, which
also modifies the oscillator’s kinetic and potential components [24]. By
examining the harmonic oscillator in curved space-time, scientists can probe
the limits of quantum mechanics in extreme environments, offering insights
into potential quantum gravitational effects and advancing the broader goal of
developing a unified theory of quantum gravity [25]. In addition to bridging
the gap between General Relativity and Quantum Field Theory, this branch of
study has applications in understanding astrophysical processes and particle
behavior in the intense gravitational fields of black holes [26] and neutron
stars [27].

In General Relativity, numerous modified theories of gravity have been
proposed to address issues that cannot be explained by GR alone. Among these
theories, Eddington-inspired Born-Infeld (EiBI) gravity [28, 29] stands out as
a fascinating and innovative theoretical framework. EiBI gravity seeks to
address some longstanding puzzles in gravitational physics by merging concepts
from two influential approaches: the Eddington gravitational theory [30] and
the Born-Infeld theory of electromagnetism. Originally proposed by M. Baños et
al. in the context of non-linear electrodynamics [31], the Born-Infeld theory
aimed to eliminate the singularities associated with point charges [32]. When
applied to gravity, this idea offers a non-linear modification to General
Relativity (GR) [33], which can potentially regularize the singularities
inherent in standard GR solutions, such as the Big Bang singularity [34] and
black hole singularities [35]. EiBI gravity posits that the space-time metric
and the connection are independent variables [36], an idea borrowed from
Eddington’s 1924 affine gravity theory [37]. The simplest solution ever
obtained in EiBI gravity is the Global Monopole/Wormhole (GM/WH) [38, 39].
This solution interpolates between a modified global monopole [40] and a
wormhole similar to the Ellis–Bronnikov wormhole with topological charge. It
is noteworthy that this solution was obtained with a source of matter that
does not violate the energy conditions and, impressively, it is as simple as
the well-known Ellis-Bronnikov solution [41, 42].

The implications of the harmonic oscillator (HO) for fundamental physics have
drawn considerable interest. Numerous scenarios have been studied for HO
solutions, providing insights into the behavior of fields and matter in curved
space-times. The HO problem has been examined in various curved space-time
backgrounds, such as in a space with a combined linear topological defect-
namely, disclination plus a dislocation [43], the topological Aharonov-Bohm
effect in a 2D scenario [44], topologically charged Ellis-Bronnikov-type
wormholes [45], cosmic string space-time with dislocation under a repulsive
inverse square potential and rotational frame effects [46], in the background
of a circularly symmetric, static wormhole space-time with cosmic strings
[47], Morris-Thorne-type wormhole backgrounds with cosmic strings [48], Born-
Infeld space-time background under the Aharonov–Bohm geometric phase [49], a
space-time with screw dislocation background under rotational and inverse-
square potential effects in the presence of a flux field [50], a space-time
with distortions such as a vertical spiral [51], an elastic medium with a
spiral dislocation [52], the effect of singular potentials [53], and conical
singularities [54, 55], influence of spiral dislocation topology on the
revival time [56], effects of a screw dislocation and a linear potential [57],
non-inertial effects in the presence of a screw dislocation [58].

Our primary objective is to explore the quantum dynamics of a harmonic
oscillator system within the context of a topologically charged Eddington-
inspired Born-Infeld (EiBI) gravity space-time. Additionally, we incorporate
the effects of a Wu-Yang magnetic monopole (WYMM) into the harmonic oscillator
system and perform a comprehensive analysis of the eigenvalue solutions.
Specifically, we demonstrate that the energy spectrum and wave function of the
harmonic oscillator are influenced by the global monopole (GM) and EiBI
gravity parameters. In addition, the presence of a WYMM further shifts the
energy spectrum and alters the wave function of the harmonic oscillator.
Moreover, we examine the harmonic oscillator system within a point-like defect
geometry, considering the influences of a WYMM and a repulsive inverse square
potential. We illustrate how these parameters affect the energy spectrum and
the radial wave function. It is shown that the eigenvalue solutions of the
harmonic oscillator system gets modification by these parameters. The point-
like global monopole space-time in spherical coordinates
$(t,r,\theta,\varphi)$ is described by the following line-element [40]

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\alpha^{2}}+r^{2}\,(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (1)

where $\alpha=(1-8\,\pi\,G\,\zeta^{2})$ is the global monopole (GM) parameter
and $\zeta$ is the energy scale of the symmetry breaking.

This paper is organized as: In Section 2, we investigate the harmonic
oscillator problem within the Eddington-inspired Born-Infeld (EiBI) gravity in
global monopole (GM) space-time. In Section 3, we consider the harmonic
oscillator problem without EiBI gravity in global monopole space-time but in
the presence of a Wu-Yang magnetic monopole (WYMM) and an external inverse
square potential, and analyze the results. In Section 4, we study the harmonic
oscillator problem in GM space-time within EiBI gravity and in the presence of
a WYMM. In Section 5, we present the conclusions. Throughout the article, we
choose the system of units, where $c=1=\hbar=G$.

##  2 Harmonic oscillator in topologically charged EiBI-gravity space-time

In this section, we study the harmonic oscillator problem within the
topologically charged Eddington-inspired Born-Infeld (EiBI) gravity space-time
and analyze the results.

Therefore, we begin this section by writing the line-element describing this
topologically charged EiBI-gravity space-time in spherical coordinates
$(t,r,\theta,\varphi)$ given by [59, 60]

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\alpha^{2}\,\left(1+\frac{\kappa}{r^{2}}\right)}+r^{2}\,d\Omega^{2},$
(2)

where $d\Omega^{2}=(d\theta^{2}+\sin^{2}\theta d\varphi^{2})$, and $\kappa$ is
the Eddington parameter that controls the non-linearity of EiBI-gravity. When
the Eddington parameter is switched off, i.e., $\kappa=0$, this metric (2)
reduces to GM one, Eq. (1) above. Therefore, the line-element (2) is sometimes
referred to as the modified GM or topologically charged EiBI-gravity space-
time. One of the most intriguing features of EiBI- gravity is its ability to
avoid cosmological singularities, resulting in entirely singularity-free
states [61, 62]. When $\alpha=1$ and $\kappa=-b^{2}<0$, where $b=const$, it
corresponds to a Morris-Thorne-type wormhole space-time [48, 63, 64, 65, 66].
Furthermore, when $\kappa<0$, it describes a topologically charged wormhole
space-time [67].

Transforming to a new variable via $r=\sqrt{\rho^{2}-\kappa}$ into the above
EiBI-gravity space-time (2) results the following metric given by

$ds^{2}=-dt^{2}+\frac{d\rho^{2}}{\alpha^{2}}+(\rho^{2}-\kappa)\,d\Omega^{2}.$
(3)

It is worthwhile mentioning that our primary interest lies in investigating
the harmonic oscillator problem in the space-time described by (3). By
expressing the above space-time in the form
$ds^{2}=-dt^{2}+g_{ij}\,dx^{i}\,dx^{j}$, we obtain the corresponding spatial
metric tensor $g_{ij}$ and its contravariant form as follows:

$g_{ij}=\begin{pmatrix}\frac{1}{\alpha^{2}}&0&0\\\ 0&(\rho^{2}-\kappa)&0\\\
0&0&(\rho^{2}-\kappa)\,\sin^{2}\theta\end{pmatrix},\quad
g^{ij}=\begin{pmatrix}\alpha^{2}&0&0\\\ 0&\frac{1}{(\rho^{2}-\kappa)}&0\\\
0&0&\frac{1}{(\rho^{2}-\kappa)\,\sin^{2}\theta}\end{pmatrix}.$ (4)

Its determinant is given by

$det(g_{ij})=g=\frac{(\rho^{2}-\kappa)^{2}\,\sin^{2}\theta}{\alpha^{2}}.$ (5)

Now, we introduce the harmonic oscillator system in the presence of an
external potential $V({\bf r})$ described by the following non-relativistic
wave equation

$\left[-\frac{1}{2\,M}\,\frac{1}{\sqrt{g}}\,\partial_{i}\,\left(g^{ij}\,\sqrt{g}\,\partial_{j}\right)+\frac{1}{2}\,M\,\omega^{2}\,\rho^{2}+V({\bf
r})\right]\,\Psi({\bf r})=E\,\Psi({\bf r}),$ (6)

where $M$ and $E$ are the particles’ mass and energy, respectively, $\omega$
is the frequency of the harmonic oscillator system, $g$ is the determinant of
the spatial metric tensor $g_{ij}$, and $\Psi({\bf r})$ is the wave function.

Form the spatial metric tensor $g_{ij}$ given in Eq. (4), it is clear that
this metric tensor depends on $(\rho,\theta)$ while being independent of the
coordinate $(\varphi)$. Let us consider the wave function ansatz $\Psi({\bf
r})$ in the following form:

$\Psi({\bf r})=R(\rho)\,Y_{\ell,m}(\theta,\varphi),$ (7)

where $R(\rho)$ is the radial wave function,
$Y_{\ell,m}(\theta,\varphi)=\Phi_{m}(\varphi)\,\Theta_{\ell,m}(\theta)$ is the
spherical harmonics, and $\ell,m$, respectively are the angular and azimuthal
quantum numbers.

Thereby expressing the wave equation (6) using (4) and the wave function
ansatz (7), we obtain the following differential equation given by

$\displaystyle\Bigg{[}\alpha^{2}\left\\{\frac{d^{2}}{d\rho^{2}}+\frac{2\,\rho}{(\rho^{2}-\kappa)}\,\frac{d}{d\rho}\right\\}+2\,M\,(E-V(\rho))-M^{2}\,\omega^{2}\,\rho^{2}-\frac{\ell\,(\ell+1)}{(\rho^{2}-\kappa)}\Bigg{]}\,R(\rho)=0,$
(8)

where we have used the following angular relations

$-\Bigg{[}\frac{1}{\sin\theta}\,\frac{d}{d\theta}\,\left(\sin\theta\,\frac{d}{d\theta}\right)+\frac{1}{\sin^{2}\theta}\,\frac{d^{2}}{d\varphi^{2}}\Bigg{]}\,Y_{\ell,m}(\theta,\varphi)=\ell\,(\ell+1)\,Y_{\ell,m}(\theta,\varphi),$
(9)

where $\ell$ is the angular quantum number which is related with the magnetic
quantum number $m$ as $\ell=(|m|+n)$ with $n=0,1,2,...$. The solutions of the
angular equations are well-known and given in many quantum mechanics
textbooks.

Now, we try to solve the above radial equation (8) and obtain analytical
solutions of the quantum system under investigation. Considering zero external
potential, $V(\rho)=0$, the radial wave equation (8) can be written as:

$\displaystyle
R^{\prime\prime}(\rho)+\frac{2\,\rho}{\rho^{2}-\kappa}\,R^{\prime}(\rho)+\Bigg{[}\Lambda-\Omega^{2}\,\rho^{2}-\frac{\tau^{2}}{\rho^{2}-\kappa}\Bigg{]}\,R(\rho)=0\,,$
(10)

where we have defined

$\Lambda=\frac{2\,M\,E}{\alpha^{2}},\quad\quad\Omega=\frac{M\,\omega}{\alpha},\quad\quad\tau^{2}=\frac{\ell\,(\ell+1)}{\alpha^{2}}\,.$
(11)

Transforming to a new function via

$R(\rho)=\exp\left(-\frac{1}{2}\,\Omega\,\rho^{2}\right)\,\psi(\rho)$ (12)

into the radial Eq. (10) results the following differential equation form:

$\displaystyle\psi^{\prime\prime}(\rho)+\Bigg{[}\frac{2\,\rho}{\rho^{2}-\kappa}-2\,\Omega\,\rho\Bigg{]}\,\psi^{\prime}(\rho)+\Bigg{[}\Delta-\frac{2\,\Omega\,\rho^{2}}{\rho^{2}-\kappa}-\frac{\tau^{2}}{\rho^{2}-\kappa}\Bigg{]}\,\psi(\rho)=0,$
(13)

where we set the parameter

$\Delta=\Lambda-\Omega=\frac{2\,M\,E}{\alpha^{2}}-\Omega.$ (14)

Finally transforming to a new variable via $\rho^{2}=s\,\kappa$ into the
equation (13) results the following second-order differential equation form
given by

$\displaystyle\psi^{\prime\prime}(s)+\left[-\Omega\,\kappa+\frac{1/2}{s}+\frac{1}{s-1}\right]\,\psi^{\prime}(s)+\Bigg{[}\frac{\left(\Delta\,\kappa+\tau^{2}\right)/4}{s}+\frac{-\left(2\,\Omega\,\kappa+\tau^{2}\right)/4}{s-1}\Bigg{]}\,\psi(s)=0.$
(15)

Equation (15) is the confluent Heun differential equation form [68, 69].
Comparing Eq. (15) with the Eq. (A1) in appendix, we obtain

$\Xi=-\Omega\,\kappa,\quad\quad\beta=-1/2,\quad\quad\gamma=0,\quad\quad\mu=\left(\Delta\,\kappa+\tau^{2}\right)/4,\quad\quad\nu=-\left(2\,\Omega\,\kappa+\tau^{2}\right)/4.$
(16)

Thus, $\psi(s)$ in Eq. (15) is the confluent Heun function given by

$\psi(s)=H_{c}\left(-\Omega\,\kappa\,,-\frac{1}{2}\,,0\,,\frac{M\,E\,\kappa}{2\,\alpha^{2}}\,,\frac{1}{4}-\frac{(2\,M\,E\,\kappa+\ell\,(\ell+1))}{4\,\alpha^{2}}\,;s\right).$
(17)

In order to solve Eq. (15), we consider a power series expansion given in the
form $H_{c}(s)=\sum^{\infty}_{i=0}\,d_{i}\,s^{i}$ [70]. Thereby, substituting
this power series into the Eq. (15), we obtain following recurrence relation

$\displaystyle
d_{i+2}=\frac{1}{(i+2)(2\,i+3)}\Big{[}\Big{\\{}(i+1)(2\,i+3+2\,\Omega\,\kappa)-2\,\mu\Big{\\}}\,d_{i+1}+2\,(-\Omega\,\kappa\,i+\mu+\nu)\,d_{i}\Big{]}$
(18)

with a few coefficients given by

$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle-2\,\mu\,d_{0},$
$\displaystyle d_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\Big{[}\Big{\\{}(3+2\,\Omega\,\kappa)-2\,\mu\Big{\\}}\,d_{1}+2\,(\mu+\nu)\,d_{0}\Big{]}.$
(19)

To find solutions for the quantum system under investigation, it is necessary
to truncate the power series $H_{c}(s)$ to a finite-degree polynomial of order
$n$ so that the radial wave function (12) remains regular everywhere. This
truncation is achieved by ensuring the coefficient $d_{n+1}=0$ in the
recurrence relation (18), where $i=(n-1)$. Consequently, the recurrence
relation Eq. (18) under this condition can be rewritten as follows:

$\displaystyle
d_{n}=-\frac{2\,\Big{[}-\Omega\,\kappa\,(n-1)+\mu+\nu\Big{]}}{\Big{[}n\,(2\,n+1+2\,\Omega\,\kappa)-2\,\mu\Big{]}}\,d_{n-1},$
(20)

where $n=1,2,3,...$ represent the radial modes of the system. Analyzing Eq.
(20) requires imposing specific values for the radial mode $n$, due to the
dependence of the coefficients on $d_{n}$ and $d_{n-1}$. Let us consider the
radial mode $n=1$, which represents the lowest state of the quantum system.
Substituting $n=1$ into Eq. (20), we obtain the following coefficient

$\displaystyle
d_{1}=-\frac{2\,(\mu+\nu)}{\Big{[}3+2\,\Omega\,\kappa-2\,\mu\Big{]}}\,\,d_{0}.$
(21)

Thus, by combining Eqs. (19) and (21) gives us the following expression of the
energy eigenvalues given by

$E_{1,\ell}=\frac{1}{2\,M}\Bigg{[}3\,M\,\omega\,\alpha-\frac{\ell\,(\ell+1)}{\kappa}+\frac{2\,\alpha^{2}}{\kappa}\pm\,\frac{\alpha^{2}}{\kappa}\,\sqrt{\frac{2\,\ell\,(\ell+1)}{\alpha^{2}}+4+\frac{4\,M\,\omega\,\kappa}{\alpha}\,\left(\frac{M\,\omega\,\kappa}{\alpha}+4\right)}\Bigg{]}.$
(22)

Equation (22) represents the allowed energy values for the lowest energy state
of the harmonic oscillator system in topologically charged EiBI-gravity space-
time. Notably, the energy spectrum of the harmonic oscillator cannot be
determined by a closed expression. On the contrary, it is only possible to
determine the allowed energy values for the quantum system by separately
imposing values of $n$ into the recurrence relation Eq. (20).

Furthermore, unlike in Ref. [71], the lowest energy state of the system is not
defined by the radial mode $n=0$ but by the radial mode $n=1$. Additionally,
it is evident that the allowed energy values for the lowest energy state of
the harmonic oscillator depends on the topological and Eddington parameters,
$(\alpha,\kappa)$. Moreover, the ground state energy value is influenced by
the oscillator frequency $\omega$ and the angular quantum number $\ell$.

The eigenfunction corresponding to the ground state of the energy spectrum
given by Eq. (22) is described by the first term of the polynomial solution to
the confluent Heun equation, Eq. (17). This eigenfunction is defined as
$\psi_{1,\ell}(s)=(d_{0}+d_{1}\,s)$. Here $d_{0}$ and $d_{1}$ are the
coefficients determined by the boundary conditions and the specifics of the
quantum system under consideration. This form represents the simplest non-
trivial solution, encapsulating the fundamental characteristics of the ground
state within the given gravitational framework. Therefore, the ground state
wave function is given by

$R_{1,\ell}(\rho)=\exp\left(-\frac{M\,\omega}{2\,\alpha}\,\rho^{2}\right)\,\Bigg{[}1-\rho^{2}\,\left\\{\frac{1}{\kappa}+\frac{M\,\omega}{\alpha}\pm\frac{1}{2\,\kappa}\,\sqrt{\frac{2\,\ell\,(\ell+1)}{\alpha^{2}}+4+\frac{4\,M\,\omega\,\kappa}{\alpha}\,\left(\frac{M\,\omega\,\kappa}{\alpha}+4\right)}\right\\}\Bigg{]}\,d_{0}.$
(23)

We observe that the ground state wave function of the harmonic oscillator
depends on the topological and Eddington parameters, $(\alpha,\kappa)$.
Additionally, it is influenced by the oscillator frequency $\omega$ and the
angular quantum number $\ell$. This dependence reflects the complex interplay
between the gravitational effects and the intrinsic characteristics of the
harmonic oscillator system in the modified space-time described by the given
metric (3).

Now, we discuss a special case corresponds to zero oscillator frequency,
$\omega=0$. Thus, by making $\omega=0$ in the energy spectrum Eq. (22), we
obtain the following expression of the allowed values of the lowest energy
state given by

$E_{1,\ell}=\frac{1}{2\,M}\Bigg{[}-\frac{\ell\,(\ell+1)}{\kappa}+\frac{2\,\alpha^{2}}{\kappa}\pm\,\frac{\alpha^{2}}{\kappa}\,\sqrt{\frac{2\,\ell\,(\ell+1)}{\alpha^{2}}+4}\Bigg{]}.$
(24)

That is, the allowed energy values for the lowest energy state of a non-
relativistic quantum particle in topologically charged EiBI-gravity space-time
are discrete. This indicates that, even in the absence of harmonic
interaction, the non-relativistic quantum particle continues to exhibit
discrete energy levels. This characteristic confinement arises from the
gravitational effects produced by the topological space-time within the
framework of EiBI-gravity.

The corresponding eigenfunction in this special case using Eq. (24) can be
expressed as

$R_{1,\ell}(\rho)=\Bigg{[}1-\rho^{2}\,\left\\{\frac{1}{\kappa}\pm\frac{1}{2\,\kappa}\,\sqrt{\frac{2\,\ell\,(\ell+1)}{\alpha^{2}}+4}\right\\}\Bigg{]}\,d_{0}.$
(25)

(a)

(b)

(c)

(d)

Figure 1: The energy level $E_{1,\ell}$ of Eq. (22) for $\ell=0$-state. Here,
$E^{+}$ indicates positive sign within the energy expression and $E^{-}$ that
for negative sign. Here $M=1$, $\kappa=0.5$.

(e)

(f)

(g)

(h)

Figure 2: The energy level $E_{1,\ell}$ of Eq. (22) for $\ell=1$-state. Here,
$E^{+}$ indicates positive sign within the energy expression and $E^{-}$ that
for negative sign. Here $M=1$, $\kappa=0.5$.

(a)

(b)

(c)

(d)

Figure 3: The wave function $R_{1,\ell}$ of Eq. (23) for $\ell=0$-state. Here,
$R^{+}$ indicates positive sign within the expression and $R^{-}$ that for
negative sign. Here $M=1$, $\kappa=0.5$.

(a)

(b)

(c)

(d)

Figure 4: The wave function $R_{1,\ell}$ of Eq. (23) for $\ell=1$-state. Here,
$R^{+}$ indicates positive sign within the expression and $R^{-}$ that for
negative sign.

We have generated several graphs illustrating the behavior of the ground state
energy spectrum, given by Eq. (22), for the $\ell=0$ and $\ell=1$ states. In
Figures 1(a) and 1(b) for the $\ell=0$ states, we observe that the energy
level linearly increases with the oscillator frequency $\omega$, and this
increase shifts upward as the topology parameter $\alpha$ increases. In
Figures 1(c) and 1(d), the energy level gradually increases with the topology
parameter $\alpha$, and this level shifts upward as the oscillator frequency
$\omega$ increases. A similar pattern is observed for the $\ell=1$ state in
Figure 2. Additionally, Figures 3 and 4 depict the behavior of the ground
state wave function (23) for these two states, $\ell=0$ and $\ell=1$,
respectively.

##  3 Harmonic oscillator system in topologically charged space-time and a
WYMM without EiBI-gravity

In this part, we study the harmonic oscillator system in topologically charged
space-time without Eddington parameter, $\kappa=0$ in the presence of a Wu-
Yang magnetic monopole (WYMM). In this WYMM environment, the electromagnetic
vector potential $\vec{A}$ is introduced into the non-relativistic wave
equation through minimal substitution
$\mathcal{D}_{i}\equiv(\partial_{i}-i\,e\,A_{i})$. Consequently, the wave
equation describing the harmonic oscillator system in curved space is given
by:

$\left[-\frac{1}{2\,M}\,\frac{1}{\sqrt{g}}\,\mathcal{D}_{i}\,\left(g^{ij}\,\sqrt{g}\,\mathcal{D}_{j}\right)+\frac{1}{2}\,M\,\omega^{2}\,\rho^{2}+V({\bf
r})\right]\,\Psi({\bf r})=E\,\Psi({\bf r}),$ (26)

where $e$ is the electric charge and $E$ is the particle’s energy.

In Ref. [73], the most elegant formalism to describe an Abelian point-like
magnetic monopole were presented. It is shown that the electromagnetic vector
potential can be described by a singularity-free expression. The
electromagnetic vector potential, defined by $A_{\mu}=(-A_{0},\vec{A})$, is
expressed in two overlapping regions, $R_{a}$ and $R_{b}$, which together
cover the entire space. Using spherical coordinate system, with the monopole
located at the origin, the vector potential in these regions is given by:

$\displaystyle R_{a}$ $\displaystyle:0\leq\theta<\frac{\pi}{2}+\delta,$
$\displaystyle r>0,\quad 0\leq\phi<2\,\pi,$ $\displaystyle R_{b}$
$\displaystyle:\frac{\pi}{2}-\delta<\theta\leq\pi,$ $\displaystyle r>0,\quad
0\leq\phi<2\,\pi,$ $\displaystyle R_{ab}$
$\displaystyle:\frac{\pi}{2}-\delta<\theta<\frac{\pi}{2}+\delta,$
$\displaystyle r>0,\quad 0\leq\phi<2\,\pi,$ (27)

where $0<\delta\leq\pi/2$ and $R_{ab}$ being the overlapping region.

The only non vanishing components of vector potential are

$\displaystyle(A_{\phi})_{a}$ $\displaystyle=g\,(1-\cos\theta),$
$\displaystyle(A_{\phi})_{b}$ $\displaystyle=-g\,(1+\cos\theta),$ (28)

$g$ being the monopole strength. In the overlapping region the non vanishing
components are related by a gauge transformation

$(A_{\phi})_{a}=(A_{\phi})_{a}+\frac{i}{e}\,S\,\partial_{\phi}\,S^{-1},\quad
S=e^{2\,i\,\sigma\,\phi}.$ (29)

In order to have single valued gauge transformation, we must have
$\sigma=e\,g$. This is known as the Dirac quantization condition.

According to the analysis by Wu and Yang in Ref. [74], the solutions of the
non-relativistic wave equation in the presence of such a vector potential are
not ordinary functions but rather sections. Specifically, the solution assumes
the values $\Psi_{a}$ and $\Psi_{b}$ in $R_{a}$ and $R_{b}$, respectively, and
satisfies the gauge transformation:

$\Psi_{a}=S\,\Psi_{b}\,.$ (30)

This gauge transformation ensures the consistency of the wave function in the
overlap region $R_{ab}$ between $R_{a}$ and $R_{b}$, reflecting the underlying
topological nature of the magnetic monopole. Thus, the wave function $\Psi$ in
the presence of the Wu-Yang magnetic monopole is described by sections that
transform appropriately under the given gauge transformation.

Therefore, explicitly writing the harmonic oscillator equation (26) in
topologically charged space-time (1) given by

$\displaystyle\frac{r^{2}}{R(r)}\,\Bigg{[}\alpha^{2}\left(\frac{d^{2}}{dr^{2}}+\frac{2}{r}\,\frac{d}{dr}\right)\,R(r)+\Big{\\{}2\,M\,(E-V(r))-M^{2}\,\omega^{2}\,r^{2}\Big{\\}}\,R(r)\Bigg{]}$
$\displaystyle=-\frac{1}{Y_{\ell^{\prime},m^{\prime}}(\theta,\varphi)}\,\Bigg{[}\frac{1}{\sin\theta}\,\frac{d}{d\theta}\,\left(\sin\theta\,\frac{d\,Y_{\ell^{\prime},m^{\prime}}(\theta,\varphi)}{d\theta}\right)+\frac{1}{\sin^{2}\theta}\,\left(\frac{d}{d\varphi}-A_{\varphi}\right)^{2}\,\,Y_{\ell^{\prime},m^{\prime}}(\theta,\varphi)\Bigg{]}.$
(31)

Here $Y_{\ell^{\prime},m^{\prime}}$ is called the conical monopole harmonics
and $\ell^{\prime},m^{\prime}$ are the modified quantum numbers due to the
presence of WYMM.

Writing the separation constant $\lambda^{\prime}$, we can write the angular
equations as follows:

$\displaystyle\Bigg{[}\frac{1}{\sin\theta}\,\frac{d}{d\theta}\,\left(\sin\theta\,\frac{d}{d\theta}\right)+\frac{1}{\sin^{2}\theta}\,\left(\frac{d}{d\varphi}-A_{\phi}\right)^{2}\Bigg{]}\,Y_{\ell^{\prime},m^{\prime}}(\theta,\varphi)=-\lambda^{\prime}\,Y_{\ell^{\prime},m^{\prime}}(\theta,\varphi),$
(32)

where the eigenvalue $\lambda^{\prime}$ is given by [75, 76, 77]

$\lambda^{\prime}=\ell^{\prime}\,(\ell^{\prime}+1),\quad\ell^{\prime}=-\frac{1}{2}+\sqrt{\frac{1}{4}+\ell\,(\ell+1)-\sigma^{2}},\quad\sigma=e\,g,\quad
m^{\prime}=m+s\,q\quad(s=\pm\,1).$ (33)

Notice that for zero electromagnetic potential $A_{\phi}=0$, there is no WYMM,
and hence, one will get back the eigenvalue of the angular equations
$\lambda=\ell\,(\ell+1)$, where $\ell=(|m|+n)\quad(n=0,1,2...)$ which is given
in quantum mechanics text books. For details the above angular eigenvalue
solution, readers are suggested to see reference [75, 76, 77].

With these, we can write the radial equation for the harmonic oscillator
system in topologically charged space-time and in a WYMM without EiBI-gravity
using Eq. (31) as follows:

$\displaystyle
R^{\prime\prime}(r)+\frac{2}{r}\,R^{\prime}(r)+\frac{1}{\alpha^{2}}\,\Bigg{[}2\,M\,(E-V(r))-M^{2}\,\omega^{2}\,r^{2}-\frac{\lambda^{\prime}}{r^{2}}\Bigg{]}\,R(r)=0.$
(34)

Below, we solve the radial equation (34) for two scenarios: the first scenario
without any external potential, $V({\bf r})=0$, and the second scenario with
an external potential of the form, $V({\bf r})=V(r)\propto\frac{1}{r^{2}}$.

###  3.1 Zero external potential, V(r)=0.

In this part, we first consider zero external potential, $V(r)=0$ into the
harmonic oscillator system under investigations. Therefore, the radial
equation (34) becomes

$\displaystyle
R^{\prime\prime}(r)+\frac{2}{r}\,R^{\prime}(r)+\frac{1}{\alpha^{2}}\,\Bigg{[}2\,M\,E-M^{2}\,\omega^{2}\,r^{2}-\frac{\lambda^{\prime}}{r^{2}}\Bigg{]}\,R(r)=0.$
(35)

Transforming first $R(r)=\frac{\psi(r)}{\sqrt{r}}$, and then to a new variable
$s=\Omega\,r^{2}$ into the Eq. (35), we obtain the following differential
equation

$\psi^{\prime\prime}(s)+\frac{1}{s}\,\psi^{\prime}(s)+\Bigg{[}-\frac{j^{2}/4}{s^{2}}-\frac{1}{4}+\frac{\Lambda}{4\,\Omega\,s}\Bigg{]}\,\psi(s)=0,$
(36)

where

$\Lambda=\frac{2\,M\,E}{\alpha^{2}},\quad
j=\sqrt{\frac{\lambda^{\prime}}{\alpha^{2}}+\frac{1}{4}}=\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}}{\alpha^{2}}+\frac{1}{4}}.$
(37)

In quantum systems, the requirement of the wave function states that it must
be finite and regular everywhere at $r\to 0$ and at $r\to\infty$. Suppose, a
possible solution to the Eq. (36) is given by the following form:

$\psi(s)=s^{j/2}\,\exp(-s/2)\,F(s),$ (38)

where $F(s)$ is an unknown function.

Substituting this solution (38) into the Eq. (36) results the following
second-order differential equation form:

$s\,F^{\prime\prime}(s)+(j+1-s)\,F^{\prime}(s)-\left(j+\frac{1}{2}-\frac{\Lambda}{4\,\Omega}\right)\,F(s)=0.$
(39)

This equation (39) is the confluent hypergeometric equation form [78], which
is a second order linear homogeneous differential equation. Therefore, its
regular solution is given by

$F(s)={}_{1}F_{1}(\mathrm{a},\mathrm{b};s),\quad\mathrm{a}=j+\frac{1}{2}-\frac{\Lambda}{4\,\Omega},\quad\mathrm{b}=j+1,$
(40)

Due to the asymptotic behavior of this solution, it is necessary that the
hypergeometric function be a polynomial function of degree $n$, which means
that the parameter $\mathrm{a}$ should be a negative integer. This condition
implies that

$\left(j+\frac{1}{2}-\frac{\Lambda}{4\,\Omega}\right)=-n\quad\quad(n=0,1,2,....).$
(41)

Simplification of the above relation results the non-relativistic energy
eigenvalue given by

$E_{n,\ell,\sigma}=2\,\left(n+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}}{\alpha^{2}}+\frac{1}{4}}+\frac{1}{2}\right)\,\alpha\,\omega.$
(42)

The corresponding wave function will be

$\displaystyle\psi_{n,\ell,\sigma}(s)$ $\displaystyle=$ $\displaystyle
s^{\frac{1}{2}\,\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}}{\alpha^{2}}+\frac{1}{4}}}\,\exp(-s/2)\,{}_{1}F_{1}\left(-n,1+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}}{\alpha^{2}}+\frac{1}{4}};s\right),$
$\displaystyle R_{n,\ell,\sigma}(s)$ $\displaystyle=$
$\displaystyle\Omega^{1/4}\,s^{\frac{1}{2}\,\left(-\frac{1}{2}+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}}{\alpha^{2}}+\frac{1}{4}}\right)}\,\exp(-s/2)\,{}_{1}F_{1}\left(-n,1+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}}{\alpha^{2}}+\frac{1}{4}};s\right)$
(43)

Equation (42) represents the energy spectrum, and Eq. (43) describes the
corresponding radial wave function of a harmonic oscillator system in
topologically charged space-time and in a Wu-Yang magnetic monopole without
EiBI-gravity effects. It is evident that the eigenvalue solution of the
harmonic oscillator system is influenced by several factors, including the
strength of the WYMM ($\sigma$), the topological parameter $\alpha$, and the
oscillator frequency $\omega$. Furthermore, Eq. (42) provides a closed-form
expression for the energy spectrum of the harmonic oscillator, which varies
with changes in the quantum numbers $(n,\ell)$.

A special case corresponds to $\sigma=0$, that is without WYMM. Therefore, the
energy spectrum (42) and the wave function (43) reduces to

$\displaystyle E_{n,\ell}$ $\displaystyle=$ $\displaystyle
2\,\left(n+\sqrt{\frac{\ell\,(\ell+1)}{\alpha^{2}}+\frac{1}{4}}+\frac{1}{2}\right)\,\alpha\,\omega,$
$\displaystyle\psi_{n,\ell}(s)$ $\displaystyle=$ $\displaystyle
s^{\frac{1}{2}\,\sqrt{\frac{\ell\,(\ell+1)}{\alpha^{2}}+\frac{1}{4}}}\,\exp(-s/2)\,{}_{1}F_{1}\left(-n,1+\sqrt{\frac{\ell\,(\ell+1)}{\alpha^{2}}+\frac{1}{4}};s\right).$
(44)

Equation (44) is the eigenvalue solution of a harmonic oscillator in global
monopole space-time which is similar to those results obtained in Refs. [54,
55, 79].

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: The energy level $E_{1,\ell}$ of Eq. (42) for $\ell=1$-state. Here
$M=1$, $\kappa=0.5$, $\sigma=0.1$.

(a)

(b)

Figure 6: The wave function $R_{n,\ell,\sigma}$ of Eq. (43) for
$\ell=1$-state. Here $M=1=\omega$, $\sigma=0.1$.

In Eq. (42), we have seen that the energy expression of the harmonic
oscillator is in closed form. To illustrate, we have generated Figure 5 for
the $\ell=1$ state, showing the nature of this closed expression of the energy
spectrum with respect to $n$, $\alpha$, and $\omega$. In Figures 5(a) and
5(b), we plot $E_{n,\ell}$ versus $n$ for different values of the topology
parameter $\alpha$ and the oscillator frequency $\omega$. Figures 5(c) and
5(d) present the graph of $E_{n,\ell}$ versus $\alpha$ for different values of
$n$ and $\omega$. Figures 5(e) and 5(f) display the graph of $E_{n,\ell}$
versus $\omega$ for different values of $n$ and $\alpha$. In these graphs, we
observe that the energy level increases linearly with $n$, $\alpha$, and
$\omega$, with varying slopes. Additionally, this increase shifts upward as
the values of the other parameters increase. In Figure 6, we have plotted the
radial wave function as given by Eq. (43) for different values of $\alpha$ and
$n$, showing its behavior under these variations.

###  3.2 Repulsive Inverse Square Potential

In this part, we want to solve the radial equation (34) for an external
potential of the type $V(r)=\frac{\eta}{r^{2}}$ [50] known as inverse square
potential, where $\eta>0$ is a constant. So, substituting this potential into
the radial Eq. (34), we obtain the following equation:

$\displaystyle
R^{\prime\prime}(r)+\frac{2}{r}\,R^{\prime}(r)+\Bigg{[}\frac{2\,M\,E}{\alpha^{2}}-\Omega^{2}\,r^{2}-\frac{\lambda^{\prime}+2\,M\,\eta}{\alpha^{2}\,r^{2}}\Bigg{]}\,R(r)=0.$
(45)

Transforming first $R(r)=\frac{\psi(r)}{\sqrt{r}}$, and then to a new variable
$s=\Omega\,r^{2}$ into the Eq. (45), we obtain the following differential
equation form:

$\psi^{\prime\prime}(s)+\frac{1}{s}\,\psi^{\prime}(s)+\Bigg{[}-\frac{\tau^{2}/4}{s^{2}}-\frac{1}{4}+\frac{\Lambda}{4\,\Omega\,s}\Bigg{]}\,\psi(s)=0,$
(46)

where $\Lambda$ is defined earlier and

$\tau=\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}+2\,M\,\eta}{\alpha^{2}}+\frac{1}{4}}.$
(47)

Analogue to the previous analysis, suppose a possible solution to the Eq. (46)
is of the following form

$\psi(s)=s^{\tau/2}\,\exp(-s/2)\,F(s),$ (48)

where $F(s)$ is an unknown function.

Substituting this solution (48) into the Eq. (46) results the following
second-order differential equation:

$s\,F^{\prime\prime}(s)+(\tau+1-s)\,F^{\prime}(s)-\left(\tau+\frac{1}{2}-\frac{\Lambda}{4\,\Omega}\right)\,F(s)=0.$
(49)

The above differential equation is the confluent hypergeometric equation form
[78]. Therefore, its regular solution is given by

$F(s)={}_{1}F_{1}(\mathrm{p},\mathrm{q};s),\quad\quad\mathrm{p}=\tau+\frac{1}{2}-\frac{\Lambda}{4\,\Omega},\quad\quad\mathrm{q}=\tau+1,$
(50)

Due to the asymptotic behavior of this solution, it is necessary that the
hypergeometric function be a polynomial function of degree $n$, which means
that the parameter $\mathrm{p}$ should be a negative integer. This condition
implies that

$\left(\tau+\frac{1}{2}-\frac{\Lambda}{4\,\Omega}\right)=-n\quad\quad(n=0,1,2,....).$
(51)

Simplification of the above relation results the non-relativistic energy
eigenvalue of the harmonic oscillator system given by

$E_{n,\ell,\sigma}=2\,\left(n+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}+2\,M\,\eta}{\alpha^{2}}+\frac{1}{4}}+\frac{1}{2}\right)\,\alpha\,\omega.$
(52)

The corresponding wave function will be

$\displaystyle\psi_{n,\ell,\sigma}(s)$ $\displaystyle=$ $\displaystyle
s^{\frac{1}{2}\,\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}+2\,M\,\eta}{\alpha^{2}}}}\,\exp(-s/2)\,{}_{1}F_{1}\left(-n,1+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}+2\,M\,\eta}{\alpha^{2}}};s\right),$
$\displaystyle R_{n,\ell,\sigma}(s)$ $\displaystyle=$
$\displaystyle\Omega^{1/4}\,s^{\frac{1}{2}\,\left(-\frac{1}{2}+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}+2\,M\,\eta}{\alpha^{2}}+\frac{1}{4}}\right)}\,\exp(-s/2)\,{}_{1}F_{1}\left(-n,1+\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}+2\,M\,\eta}{\alpha^{2}}+\frac{1}{4}};s\right).$
(53)

Equation (52) is the non-relativistic energy spectrum, while (53) describes
the corresponding radial wave function of a harmonic oscillator in the context
of global monopole (GM) space-time and in a WYMM without EiBI-gravity effects,
under the influence of an external inverse square potential. The eigenvalue
solution of the harmonic oscillator system is influenced by several factors,
including the strength of the WYMM ($\sigma$), the topological parameter
($\alpha$), the oscillator frequency ($\omega$), and the potential parameter
($\eta$). Additionally, Eq. (52) provides a closed-form expression for the
energy spectrum of the harmonic oscillator, which varies with the quantum
numbers $\\{n,\ell\\}$.

A special case corresponds to $\sigma=0$, which implies the absence of WYMM.
It means the harmonic oscillator system under the influence of an inverse
square potential in GM space-time, without EiBI-gravity and WYMM effects. In
other words, it describes the non-relativistic quantum particles interacting
with a pseudoharmonic-type potential in a GM space-time background, devoid of
additional effects such as EiBI-gravity and WYMM. Consequently, the non-
relativistic energy spectrum given by Eq. (52) and the wave function described
by Eq. (53) reduce to the following form:

$\displaystyle E_{n,\ell}$ $\displaystyle=$ $\displaystyle
2\,\left(n+\sqrt{\frac{\ell\,(\ell+1)+2\,M\,\eta}{\alpha^{2}}+\frac{1}{4}}+\frac{1}{2}\right)\,\alpha\,\omega\,,$
$\displaystyle\psi_{n,\ell}(s)$ $\displaystyle=$ $\displaystyle
s^{\frac{1}{2}\,\sqrt{\frac{\ell\,(\ell+1)+2\,M\,\eta}{\alpha^{2}}}}\,\exp(-s/2)\,{}_{1}F_{1}\left(-n,1+\sqrt{\frac{\ell\,(\ell+1)+2\,M\,\eta}{\alpha^{2}}};s\right)\,.$
(54)

Equation (54) represents the non-relativistic energy spectrum of a harmonic
oscillator influenced by an inverse square potential in global monopole (GM)
space-time, excluding the effects of EiBI-gravity and the Weinberg-Yang-Mills
model (WYMM). This result is consistent with those obtained in Ref. [79].
Consequently, it is clear that the presence of a WYMM in the harmonic
oscillator system significantly affects both the energy spectrum (52) and the
wave function (53), alongside other parameters considered in this study.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 7: The energy level $E_{n,\ell,\sigma}$ of Eq. (52) for $\ell=1$-state.
Here $M=1$, $\kappa=0.5$, $\sigma=0.1$.

(a)

(b)

(c)

(d)

Figure 8: The wave function $R_{n,\ell,\sigma}$ of Eq. (53) for
$\ell=0$-state. $M=1=\omega$, $\sigma=0.1$.

(e)

(f)

(g)

(h)

Figure 9: The wave function $R_{n,\ell,\sigma}$ of Eq. (53) for
$\ell=1$-state. $M=1=\omega$, $\sigma=0.1$.

In Eq. (52), the energy spectrum is expressed in a closed or compact form and
varies with the quantum numbers $(n,\ell)$. For $\ell=1$-state solution, we
have generated Figures 7(a) through 7(i) to illustrate this energy spectrum as
functions of $n$, $\alpha$, and $\omega$. We observe that the energy spectrum
for a particular state linearly increases with different slopes when plotted
against these parameters. Additionally, as the values of the other parameters
increase, this linear increase shifts the energy level upward. While we have
focused on the $\ell=1$ state, one could also explore the $\ell=0$ state,
which we have left for future investigation.

In Figures 8 and 9, we have plotted the radial wave function as given by Eq.
(53), varying the values of different parameters involved for both the
$\ell=0$ and $\ell=1$-states. These plots demonstrate the behavior of the
radial wave function under various conditions.

##  4 Harmonic oscillator system in topologically charged EiBI-gravity space-
time and in a WYMM

In this section, we now study the harmonic oscillator system in topologically
charged space-time within the context of EiBI-gravity and in a WYMM.
Therefore, the radial equation describing harmonic oscillator system in
absence of external potential $V(\rho)=0$ in this environment is given by

$\displaystyle\Bigg{[}\frac{d^{2}}{d\rho^{2}}+\frac{2\,\rho}{(\rho^{2}-\kappa)}\,\frac{d}{d\rho}+\frac{2\,M\,E}{\alpha^{2}}-\frac{M^{2}\,\omega^{2}}{\alpha^{2}}\,\rho^{2}-\frac{\lambda^{\prime}/\alpha^{2}}{(\rho^{2}-\kappa)}\Bigg{]}\,R(\rho)=0,$
(55)

Transforming to a new function defined in Eq. (12) into the Eq. (55) results

$\displaystyle\psi^{\prime\prime}(\rho)+\Bigg{[}\frac{2\,\rho}{\rho^{2}-\kappa}-2\,\Omega\,\rho\Bigg{]}\,\psi^{\prime}(\rho)+\Bigg{[}\Delta-\frac{2\,\Omega\,\rho^{2}}{\rho^{2}-\kappa}-\frac{\iota^{2}}{\rho^{2}-\kappa}\Bigg{]}\,\psi(\rho)=0,$
(56)

where $\Delta,\Omega$ are given in Eq. (14) and

$\iota=\sqrt{\frac{\ell\,(\ell+1)-\sigma^{2}}{\alpha^{2}}}.$ (57)

Finally transforming to a new variable via $\rho^{2}=s\,\kappa$ into the
equation (56) results the following differential equation form

$\displaystyle\psi^{\prime\prime}(s)+\left[-\Omega\,\kappa+\frac{1/2}{s}+\frac{1}{s-1}\right]\,\psi^{\prime}(s)+\Bigg{[}\frac{\left(\Delta\,\kappa+\iota^{2}\right)/4}{s}+\frac{-\left(2\,\Omega\,\kappa+\iota^{2}\right)/4}{s-1}\Bigg{]}\,\psi(s)=0.$
(58)

Comparing Eq. (58) with Eq. (A1) in appendix, we obtain different parameters

$\Xi=-\Omega\,\kappa,\quad\quad\beta=-1/2,\quad\quad\gamma=0,\quad\quad\tilde{\mu}=\left(\Delta\,\kappa+\iota^{2}\right)/4,\quad\quad\tilde{\nu}=-\left(2\,\Omega\,\kappa+\iota^{2}\right)/4.$
(59)

Thus, $\psi(s)$ is the confluent Heun function given by

$\psi(s)=H_{c}\left(-\Omega\,\kappa,-\frac{1}{2},0,\frac{M\,E\,\kappa}{2\,\alpha^{2}},\frac{1}{4}-\frac{1}{4\,\alpha^{2}}\left(2\,M\,E\,\kappa+\ell\,(\ell+1)-\sigma^{2}\right);s\right).$
(60)

In order to solve q. (58), we consider a power series expansion given by
$H_{c}(s)=\sum^{\infty}_{i=0}\,d_{i}\,s^{i}$ [70]. Substituting this power
series into the Eq. (58), we obtain following recurrence relation

$\displaystyle
d_{i+2}=\frac{1}{(i+2)(2\,i+3)}\Big{[}\Big{\\{}(i+1)(2\,i+3+2\,\Omega\,\kappa)-2\,\tilde{\mu}\Big{\\}}\,d_{i+1}+2\,(-\Omega\,\kappa\,i+\tilde{\mu}+\tilde{\nu})\,d_{i}\Big{]}$
(61)

with a few coefficients given by

$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle-2\,\tilde{\mu}\,d_{0},$
$\displaystyle d_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\Big{[}\Big{\\{}(3+2\,\Omega\,\kappa)-2\,\tilde{\mu}\Big{\\}}\,d_{1}+2\,(\tilde{\mu}+\tilde{\nu})\,d_{0}\Big{]}.$
(62)

Analogue to the previous analysis done in section 2, we now truncate the power
series to a finite degree polynomial of order $n$. This truncation is achieved
by ensuring the coefficient $d_{n+1}=0$ in the recurrence relation (61), where
$i=(n-1)$. Consequently, the recurrence relation Eq. (61) under this condition
can be rewritten as follows:

$\displaystyle
d_{n}=-\frac{2\,\Big{[}-\Omega\,\kappa\,(n-1)+\tilde{\mu}+\tilde{\nu}\Big{]}}{\Big{[}n\,(2\,n+1+2\,\Omega\,\kappa)-2\,\tilde{\mu}\Big{]}}\,d_{n-1},$
(63)

where $n=1,2,3,...$ represent the radial modes of the quantum system.

As done in the earlier section, let us consider the radial mode $n=1$, which
represents the lowest energy state of the quantum system. Therefore,
substituting $n=1$ into Eq. (63), we obtain the following expression

$\displaystyle
d_{1}=-\frac{2\,(\tilde{\mu}+\tilde{\nu})}{\Big{[}3+2\,\Omega\,\kappa-2\,\tilde{\mu}\Big{]}}\,\,d_{0}.$
(64)

Thus, by combining Eqs. (62) and (64) gives us the following expression of the
energy eigenvalues given by

$\displaystyle E_{1,\ell,\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2\,M}\Bigg{[}3\,M\,\omega\,\alpha-\frac{\ell\,(\ell+1)}{\kappa}+\frac{\sigma^{2}}{\kappa}+\frac{2\,\alpha^{2}}{\kappa}$
(65)
$\displaystyle\pm\,\frac{\alpha^{2}}{\kappa}\,\sqrt{\frac{2\,\ell\,(\ell+1)-2\,\sigma^{2}}{\alpha^{2}}+4+\frac{4\,M\,\omega\,\kappa}{\alpha}\,\left(\frac{M\,\omega\,\kappa}{\alpha}+4\right)}\Bigg{]}.$

Equation (65) represents the allowed energy values for the lowest energy state
of harmonic oscillator in topologically charged EiBI-gravity space-time and in
a WYMM.

The eigenfunction corresponds to the ground state of the energy spectrum Eq.
(65) is described by the first term of the polynomial of the confluent Heun
equation Eq. (60), defined as $\psi_{1,\ell,\sigma}(s)=(d_{0}+d_{1}\,s)$.
Therefore, the ground state wave function is given by

$\displaystyle R_{1,\ell,\sigma}(\rho)$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{M\,\omega}{2\,\alpha}\,\rho^{2}\right)\,\Bigg{[}1-\rho^{2}\,\Bigg{\\{}\frac{1}{\kappa}+\frac{M\,\omega}{\alpha}\pm\frac{1}{2\,\kappa}\times$
(66)
$\displaystyle\sqrt{\frac{2\,\ell\,(\ell+1)-2\,\sigma^{2}}{\alpha^{2}}+4+\frac{4\,M\,\omega\,\kappa}{\alpha}\,\left(\frac{M\,\omega\,\kappa}{\alpha}+4\right)}\Bigg{\\}}\Bigg{]}\,d_{0}.$

Now, we discuss a special case corresponds to zero oscillator frequency,
$\omega=0$. So, by making $\omega=0$ in the energy spectrum Eq. (65), we
obtain

$E_{1,\ell,\sigma}=\frac{1}{2\,M}\Bigg{[}-\frac{\ell\,(\ell+1)}{\kappa}+\frac{\sigma^{2}}{\kappa}+\frac{2\,\alpha^{2}}{\kappa}\pm\,\frac{\alpha^{2}}{\kappa}\,\sqrt{\frac{2\,\ell\,(\ell+1)-2\,\sigma^{2}}{\alpha^{2}}+4}\Bigg{]}.$
(67)

That is, the allowed energy values for the lowest energy state of a non-
relativistic quantum particle in topologically charged EiBI-gravity space-time
and in a WYMM. This means that, even without harmonic interaction, the non-
relativistic quantum particle continues to have discrete energy,
characterizing the confinement, which comes from the gravitational effects.
The corresponding eigenfunction in this special case from Eq. (66) reduces as
follows:

$R_{1,\ell,\sigma}(\rho)=\Bigg{[}1-\rho^{2}\,\left\\{\frac{1}{\kappa}\pm\frac{1}{2\,\kappa}\,\sqrt{\frac{2\,\ell\,(\ell+1)-2\,\sigma^{2}}{\alpha^{2}}+4}\right\\}\Bigg{]}\,d_{0}.$
(68)

(a)

(b)

(c)

(d)

Figure 10: The energy level $E_{1,\ell}$ of Eq. (65) for $\ell=0$-state. Here,
$E^{+}$ indicates positive sign within the energy expression and $E^{-}$ that
for negative sign. Here $M=1$, $\kappa=0.5$, $\sigma=0.1$.

(a)

(b)

(c)

(d)

Figure 11: The energy level $E_{1,\ell}$ of Eq. (65) for $\ell=1$-state. Here,
$E^{+}$ indicates positive sign within the energy expression and $E^{-}$ that
for negative sign. Here $M=1$, $\kappa=0.5$, $\sigma=0.1$.

(a)

(b)

(c)

(d)

Figure 12: The wave function $R_{1,\ell,\sigma}$ of Eq. (66) for
$\ell=0$-state. Here, $R^{+}$ indicates positive sign within the expression
and $R^{-}$ that for negative sign. Here $M=1$, $\kappa=0.5$, $\sigma=0.1$.

(e)

(f)

(g)

(h)

Figure 13: The wave function $R_{1,\ell,\sigma}$ of Eq. (66) for
$\ell=1$-state. Here, $R^{+}$ indicates positive sign within the expression
and $R^{-}$ that for negative sign. Here $M=1$, $\kappa=0.5$, $\sigma=0.1$.

Equation (65) describes the ground state energy level of a harmonic
oscillator, while Equation (66) provides the corresponding radial wave
function for the radial mode defined by $n=1$. We have generated Figure 10 for
the $\ell=0$ state to show the behavior of the ground state energy level with
respect to $\alpha$ and $\omega$. In Figures 10(a) and 10(b), we observe that
the energy level gradually increases. In Figures 10(c) and 10(d), the energy
level increases linearly. We have also plotted the ground state energy level
for the $\ell=1$ state in Figure 11, demonstrating its behavior with different
parameters. Similarly, we have generated Figure 12 for the $\ell=0$ state and
Figure 13 for the $\ell=1$ state, illustrating the behavior of the ground
state radial wave function. These figures show how the radial wave function
changes with different parameter values.

##  5 Conclusions

The significance of the eigenvalue solutions obtained in the context of curved
space is much greater compared to those in flat space. This is because
gravitational effects are taken into account when investigating the quantum
dynamics of particles, leading to modified results. Numerous studies have
explored the quantum motions of spin-0 and spin-1/2 particles in various
curved and conical geometries. In this paper, we aimed to study the quantum
dynamics of a harmonic oscillator system within the framework of topologically
charged Eddington-inspired Born-Infeld (EiBI) gravity space-time, in the
presence of a Wu-Yang magnetic monopole (WYMM). We analyzed how these modified
theories alter the quantum dynamics of a harmonic oscillator system.

In Section 2, we investigated the quantum dynamics of a harmonic oscillator in
a topologically charged EiBI-gravity space-time. We derived the radial
equation and obtained an analytical solution, presenting the ground state
energy level $E_{1,\ell}$ and the corresponding wave function $R_{1,\ell}$.
various parameters, such as GM parameter $\alpha$, the strength of WYMM
$\sigma$, the Born-Infeld parameter $\kappa$, and the oscillator frequency
$\omega$, alters the eigenvalue solutions of the harmonic oscillator. It is
worth noting that, unlike the results in Ref. [71], we could not derive a
closed-form expression for the energy spectrum. To illustrate the nature of
the energy levels and the radial wave function, we generated several figures
showing their behavior as various parameters change.

In Section 3, we studied the quantum dynamics of a harmonic oscillator in GM
space-time without EiBI gravity, i.e., with the EiBI parameter $\kappa=0$.
Additionally, we incorporated a WYMM into the quantum system and derived the
radial equation in the presence of an external potential $V({\bf r})$. We then
solved this radial equation under two scenarios: zero potential (Sub-section
3.1) and an inverse square potential (Sub-section 3.2). In each sub-section,
we presented analytical solutions and obtained closed-form expressions for the
energy spectrum and the radial wave functions. Our analysis showed that
various factors, such as the GM parameter $\alpha$, the strength of the WYMM
characterized by $\sigma$, the potential depth characterized by $\eta$, and
the oscillator frequency $\omega$, significantly alter the eigenvalue
solutions of the harmonic oscillator, leading to deviations from the flat
space results. To illustrate the behavior of the energy spectrum and the
radial wave function, we generated several figures showing the effects of
varying these parameters.

In Section 4, we investigated the quantum dynamics of a harmonic oscillator in
a topologically charged EiBI-gravity space-time in the presence of a Wu-Yang
magnetic monopole (WYMM) field. Following a similar procedure as in Section 2,
we presented the ground state energy level and the corresponding wave function
of the harmonic oscillator. Our study showed that various parameters, such as
the GM parameter $\alpha$, the strength of the WYMM $\sigma$, the Born-Infeld
parameter $\kappa$, and the oscillator frequency $\omega$, significantly alter
the eigenvalue solutions of the harmonic oscillator.

Our study of the quantum dynamics of a harmonic oscillator in the context of
topologically charged Born-Infeld gravity space-time sheds light on the
complex interactions between the quantum mechanical systems and gravitational
fields. We have demonstrated how the Eddington parameter influences the
behavior of the harmonic oscillator in a global monopole space-time. Our
results underscore the importance of considering nonlinear gravitational
effects in high-energy quantum systems, showing that the Eddington parameter
plays a crucial role in defining corrections to the energy levels and wave-
functions.

Additionally, we explored the behavior of the harmonic oscillator in the
absence of EiBI gravity effects ($\kappa=0$), focusing on the presence of a
WYMM and an inverse square potential. We found that the strength of the WYMM
field and the potential depth significantly alter the harmonic oscillator’s
behavior in a global monopole space-time in addition to other parameters.
Unlike the conventional quantum harmonic oscillator, the inclusion of an
inverse square potential introduces additional complexity to the system,
resulting in a richer structure of the energy spectrum and wave-functions.

## Data Availability

No data were generated or analysed in this study.

## Conflict of Interests

There is no conflicts of interests.

## Funding Statement

No fund has received for this paper.

## Acknowledgements

F.A. acknowledges the Inter University Centre for Astronomy and Astrophysics
(IUCAA), Pune, India for granting visiting associateship.

## Appendix: The Confluent Heun Equation

The standard form of the confluent Heun equation is Refs. [68, 69]

$\displaystyle
H^{\prime\prime}(x)+\Big{[}\Xi+\frac{\beta+1}{x}+\frac{\gamma+1}{x-1}\Big{]}\,H^{\prime}(x)+\Big{[}\frac{\mu}{x}+\frac{\nu}{x-1}\Big{]}\,H(x)=0,$
(A.1)

where $H(x)=H_{c}(\Xi,\beta,\gamma,\delta,\chi;x)$ is the confluent Heun
function. The parameters $\mu$ and $\nu$ given in the last term of Eq. (B.1)
are defined as

$\displaystyle\mu=\frac{1}{2}\,(\Xi+\Xi\,\beta-\beta-\beta\,\gamma-\gamma)-\chi,\quad\nu=\frac{1}{2}\,(\Xi+\Xi\,\gamma+\beta+\beta\,\gamma+\gamma)+\delta+\chi.$
(A.2)

By using the Frobenius method Refs. [70], we can obtain a polynomial solution
to the confluent Heun equation. Let us write the confluent Heun function as a
power series around the origin,

$H(x)=\sum^{\infty}_{i=0}\,d_{i}\,x^{i},$ (A.3)

where $d_{i}$ are the coefficients.

Thereby, substituting this power series in the Eq. (A.1) we obtain a few
coefficients:

$\displaystyle d_{1}=-\frac{\mu}{\beta+1}\,d_{0},$ (A.4)

with the following recurrence relation

$\displaystyle
d_{k+2}=\frac{1}{(k+2)(k+2+\beta)}\Big{[}\Big{\\{}(k+1)(k+2+\beta+\gamma-\Xi)-\mu\Big{\\}}\,d_{k+1}+(\Xi\,k+\mu+\nu)\,d_{k}\Big{]}.$
(A.5)

Therefore, from the Eq. (A.5), the confluent Heun series becomes a polynomial
of degree $n$ when we impose two conditions:

$\displaystyle
d_{n+1}=0,\quad\delta=-\Xi\,\Big{[}n+\frac{1}{2}\,(2+\beta+\gamma)\Big{]},$
(A.6)

where $n=1,2,3,....$. But, we do not know if a closed expression for the
asymptotic behaviour of the confluent Heun function for large values of its
argument exists.

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