Hydrogen and Pionic Atoms Under the Effects of Oscillations in the Global Monopole Spacetime

Abstract

In this analysis, we investigate hydrogen and pionic atoms subjected to Dirac and Klein-Gordon oscillators, respectively, in the global monopole spacetime. Through a purely analytical analysis, we determine solutions of bound state, in which we define the allowed energy values for the lowest energy state of both proposed systems. In addition to the influence of the topological defect on the results obtained, we note another quantum effect: the oscillation frequencies of both systems depend on the system quantum numbers, that is, the angular frequencies are quantized.

1. Introduction

One of the most traditional problems in the course of modern physics, in particular, quantum mechanics, is the problem of the hydrogen atom, that is, an atom with a single electron orbiting in its electrosphere [1]. This problem, in the academic literature, is treated by the Schröedinger equation, since it is a quantum system that can be solved analytically. In addition, it is a prototype for investigating heavier atoms through computational support. The hydrogen atom is also investigated in more general situations via specific wave equations, which encompass the relativistic effects contained in the atomic structure.

Among these wave equations, we have the Dirac equation and the Klein–Gordon equation, both used to deal with atoms with a single fundamental particle orbiting the atomic nucleus. With the Dirac equation, we effectively deal with the hydrogen atom (HA), while with the Klein–Gordon equation, we deal with the pionic atom (PA), in which, in both cases, it is possible to obtain the results provided by the Schröedinger equation, in the non-relativistic regime [2]. It is worth noting that HA and PA go far beyond mere academic problems, as they are fundamental physical systems capable of providing results that can be adopted approximately in more complex quantum structures, whether from a theoretical or experimental point of view, culminating in several applications in other related areas.

In Ref. [2], HA and PA are treated in a trivial flat geometry, known in the literature as Minkowski spacetime. However, these systems have been treated in more general and complex geometries, in which ingredients of a purely gravitational nature are considered. Among these gravitational ingredients, there are topological defects, objects predicted in fundamental theories about the beginnings of the Universe [3]. From a mathematical point of view, topological defects are solutions of nonlinear differential equations; from a physical point of view, they are interfaces or regions capable of separating two or more different physical states [4].

The best-known topological defects in the literature are the cosmic string [5,6,7,8], domain walls [4,9], and the global monopole (GM) [10], all associated with the curvature of spacetime [3]. It is theoretically predicted that these defects may have arisen in the early dawn of the Universe through the decoupling of fundamental interactions and that they are in regions of the Universe, despite us not yet having observational evidence for this. Among the defects mentioned, a strong candidate to be observed is the GM [3].

In particular, GM has been investigated in several branches of physics, for example, in a gravitating magnetic monopole [11], in the presence of a Wu Yang magnetic monopole [12], in $f\left(R\right)$ theory [13], on a scalar self-energy for a charged particle [14], on a massless scalar field [15], and under vacuum polarization for a massless spin-$1/2$ field [16]. Within the scope of quantum mechanics, GM has been investigated in both non-relativistic and relativistic scenarios. It is worth mentioning that in low-energy scenarios, the GM can be seen as a point-type defect, which, in crystallography, can be a vacancy or an impurity [17]. In this sense, in non-relativistic quantum mechanics scenarios, GM has been investigated via Kratzer potential [18], via a charged-particle-magnetic monopole scattering [19], on a particle under the effects of self-interaction potential [20], and on an oscillator [21,22]. In terms of relativistic cases, there are studies on scalar bosons [23]; on the presence of a dyon, magnetic flux, and scalar potential [24]; on the HA and PA [25]; and on the Dirac oscillator (DO) and Klein–Gordon oscillator (KGO) [26].

As stated above, the HA has been investigated in the presence of topological defects. For example, in Ref. [27], this quantum system is analyzed in a background characterized by the presence of the cosmic string. Another background considered in this reference is the GM spacetime. In both cases, bound state solutions are determined, in which the influence of topological defects on the energy levels of the HA is emphasized. In particular, for the GM, this influence is seen as a reduction in their allowed energy values. In Ref. [25], both HA and PA are investigated in the GM spacetime; however, they are investigated with a metric modified by parametrization, which does not modify the physics-based effects of the defect, but rather the structure of the metric, with the aim being to facilitate mathematical development. In Ref. [25], the energy levels of both systems are determined, and again, the modification of the allowed energy values was noticeable through a reduction in their values due to the presence of the topological defect.

It is well known in the literature [28] that all potential energy, around its point of stability, has a regime of low oscillations, that is, this potential energy behaves like a harmonic oscillator. This scenario has been investigated in systems of diatomic molecules described, for example, by the Morse potential [18] and, more recently, by the Krtazer–Fues potential [29], a particular case of Mie-type potential. In relativistic scenarios, the radial oscillator term has been inserted into relativistic wave equations through non-minimal couplings, such as modification of the mass term [2,30,31,32,33,34] or by non-trivial non-minimal couplings, as in the case of couplings inspired by the Standard Model Extension of particle physics [35,36,37]. This need for non-minimal couplings to insert the oscillator potential into relativistic wave equations is due to incompatibility with gauge invariance in electromagnetism via minimal coupling [38].

An interesting proposal for a relativistic oscillator model was made by Moshinsky and Szczepaniak [38]. This is a spin-1/2 fermionic relativistic oscillator model, that is, this model is introduced into the Dirac equation via non-minimal coupling and has an analytical solution. Furthermore, in the non-relativistic limit, strong spin–orbit coupling is obtained. This relativistic spin-$1/2$ oscillator model became known in the literature as the DO [38,39]. Then, Bruce and Minning, inspired by DO [38], proposed a relativistic spin-0 oscillator model through a non-minimal coupling in the Klein–Gordon equation [40]. The success of this model is due to its analyticity and the recovery of the Schröedinger oscillator in the non-relativistic limit [41]. This relativistic spin-0 oscillator model has become known in the literature as KGO [40].

Therefore, as there are no studies in the literature investigating HA and PA under the effects of oscillations in the GM spacetime, our research goal is to propose a solution in this sense, with the aim of determining solutions of bound state and describing the relativistic energy profile of these two systems under the effects of oscillations. It is worth noting that the Coulomb potential plus the oscillator potential is a particular case of the quark and antiquark interaction [31], an interaction that exists in atomic nuclei and is of great interest in fundamental physics.

The structure of this paper is as follows: In Section 2, we investigate HA under effects of DO in GM spacetime, where we determine the allowed values of relativistic energy for the ground state. In Section 3, we study PA under effects of KGO in GM spacetime, where we determine the allowed values of relativistic energy for the ground state. In Section 4, we present our conclusions.

2. HA Under Effects of DO in GM Spacetime

From now on, let us consider the HA immersed in the GM spacetime described by the following metric $(c=\hslash =1)$:

$$\begin{array}{c}\hfill d{s}^2=-d{t}^2+{\displaystyle \frac{d{r}^2}{{\alpha }^2}}+r^2(d{\theta }^2+{sin}^2\theta d{\phi }^2),\end{array}$$

(1)

with $0\le r<\infty$, $0\le \theta \le \pi$, $0\le \phi \le 2\pi$, and $\alpha =1-8{\pi }^{2}G{\eta }_0^2<1$ [25] is the parameter that characterizes the presence of the topological defects (GM) in the spacetime, which has a curvature scalar given by $R=R_{\mu }^{\mu }=2{\displaystyle \frac{(1-{\alpha }^2)}{r^2}}$ [3]. In this case, for the description of this system, not only in this background, but in any medium with non-trivial geometry, we must consider the Dirac equation in the following form:

$$\begin{array}{c}\hfill [i{\gamma }^{\mu }\left(x\right)({\partial }_{\mu }+{\Gamma }_{\mu }+ie{A}_{\mu })-m]\Psi \left(x\right)=0,\end{array}$$

(2)

where ${\gamma }^{\mu }\left(x\right)$ represents the generalized Dirac matrices for the spacetime described by Equation (1) in terms of the Dirac matrices in Minkowski spacetime ${\gamma }^{\left(a\right)}$, that is,

$$\begin{array}{c}\hfill {\gamma }^{\mu }\left(x\right)=e_{\left(a\right)}^{\mu }{\gamma }^{\left(a\right)},\end{array}$$

(3)

which obey the relation of anticomutation ${\gamma }^{\mu }\left(x\right){\gamma }^{\nu }\left(x\right)+{\gamma }^{\nu }\left(x\right){\gamma }^{\mu }\left(x\right)=2g^{\mu \nu }\left(x\right)$ and $e_{\left(a\right)}^{\mu }$ are the tetrad base components. The term ${\Gamma }_{\mu }$ is the spinorial connection component defined in terms of the Christoffel symbols and the tetrad components

$$\begin{array}{c}\hfill {\Gamma }_{\mu }={\displaystyle \frac{1}{4}}{\gamma }^{\left(a\right)}{\gamma }^{\left(b\right)}{e}_{\left(a\right)}^{\nu }({\partial }_{\mu }{e}_{\nu \left(b\right)}+{\Gamma }_{\mu \nu }^{\lambda }).\end{array}$$

(4)

It is worth mentioning that we are adopting Greek indices to denote the components or variables in curved spacetime, while latin indices are used to represent the components in Minkowski spacetime. In this sense, the metric tensor of non-trivial spacetime is related to the metric Minkowski tensor through the tetrad basis with the following relation:

$$\begin{array}{c}\hfill g^{\mu \nu }\left(x\right)=e_{\left(a\right)}^{\mu }{e}_{\left(b\right)}^{\nu }{\eta }^{ab},\end{array}$$

(5)

where, for our case, the tetrad base adopted herein is of GM spacetime [25]:

$$\begin{array}{c}\hfill e_{\left(a\right)}^{\mu }\left(x\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \alpha sin\theta cos\phi & {\displaystyle \frac{cos\theta cos\phi }{r}}& -{\displaystyle \frac{sin\phi }{rsin\theta }}\\ 0& \alpha sin\theta sin\phi & {\displaystyle \frac{cos\theta sin\phi }{r}}& {\displaystyle \frac{cos\phi }{rsin\theta }}\\ 0& \alpha cos\theta & -{\displaystyle \frac{sin\theta }{r}}& 0\end{array}\right],\end{array}$$

(6)

where $\mu ,\nu =0,1,2,3$, and $\left(a\right),\left(b\right)=0,1,2,3$. By using this representation of the tetrad base, the relations between curved and flat Dirac matrices are given as follows:

$$\begin{array}{c}\hfill {\gamma }^0={\gamma }^t;{\gamma }^1=\alpha \overrightarrow{\gamma }.\widehat{r}=\alpha {\gamma }^{\left(r\right)};{\gamma }^2={\displaystyle \frac{1}{r}}\overrightarrow{\gamma }.\widehat{\theta }={\displaystyle \frac{1}{r}}{\gamma }^{\left(\theta \right)};{\gamma }^3={\displaystyle \frac{\overrightarrow{\gamma }.\widehat{\phi }}{rsin\theta }}={\displaystyle \frac{{\gamma }^{\left(\phi \right)}}{rsin\theta }},\end{array}$$

(7)

where $\widehat{r}$, $\widehat{\theta }$, and $\widehat{\phi }$ are the spherical versors. In addition, the nonzero spinor connections are

$$\begin{array}{ccc}\hfill {\Gamma }_2& =& {\displaystyle \frac{(\alpha -1)}{2}}[{\gamma }^{\left(1\right)}{\gamma }^{\left(3\right)}cos\phi +{\gamma }^{\left(2\right)}{\gamma }^{\left(3\right)}sin\phi ];\hfill \\ \hfill {\Gamma }_3& =& -{\displaystyle \frac{(\alpha -1)}{2}}[{\gamma }^{\left(1\right)}{\gamma }^{\left(2\right)}sin\theta +{\gamma }^{\left(1\right)}{\gamma }^{\left(3\right)}cos\theta sin\phi -{\gamma }^{\left(2\right)}{\gamma }^{\left(3\right)}cot\theta cos\phi ]sin\theta ,\hfill \end{array}$$

(8)

from which we obtain the relation

$$\begin{array}{c}\hfill {\gamma }^2{\Gamma }_2+{\gamma }^3{\Gamma }_3={\displaystyle \frac{(\alpha -1)}{r}}\overrightarrow{\gamma }.\widehat{r}.\end{array}$$

(9)

With all these mathematical tools in hand, we can go back to Equation (2) and rewrite it in the form needed to describe the HA. To do this, we must impose that [2]

$$\begin{array}{c}\hfill A_{\mu }=\left(A_0=-{\displaystyle \frac{Ze}{r}},0,0,0\right),\end{array}$$

(10)

with e being the elementary electric charge and Z being the atomic number. Therefore, by substituting Equations (3)–(9) into Equation (2), we obtain

$$\begin{array}{c}\hfill \left[i{\gamma }^{\left(t\right)}({\partial }_t-ie{A}_t)+i{\gamma }^{\left(r\right)}\left(\alpha {\partial }_r+{\displaystyle \frac{(\alpha -1)}{r}}\right)+i{\displaystyle \frac{{\gamma }^{\left(\theta \right)}}{r}}{\partial }_{\theta }+i{\displaystyle \frac{{\gamma }^{\left(\phi \right)}}{rsin\theta }}{\partial }_{\phi }-m\right]\psi =0,\end{array}$$

(11)

for which we use the definitions

$$\begin{array}{c}\hfill {\gamma }^{\left(t\right)}=\left[\begin{array}{cc}{I}_2& 0\\ 0& -I_2\end{array}\right];{\gamma }^{\left(i\right)}=\left[\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right],\end{array}$$

(12)

where $I_2$ is the order two identity matrix, and ${\sigma }^i$ represents the Pauli matrices.

Equation (11) is the Dirac equation for the GM spacetime, which describes the relativistic quantum dynamics of a charged spin-$1/2$ particle subjected to the arbitrary potential $e{A}_t=V\left(\overrightarrow{r}\right)$. For our case, this potential is defined by Equation (10), which gives us the HA interaction. In addition, let us introduce the DO term through non-minimal coupling [38,42]: ${\partial }_r\to {\partial }_r+m\omega \widehat{\beta }r$, where $\omega$ is the angular frequency of DO and $\widehat{\beta }={\gamma }^{\left(0\right)}$. Before we continue developing our problem, it is important to mention that the DO has been investigated in several relativistic quantum mechanics scenarios, for example, in an anti-de Sitter spacetime [43], in a spinning cosmic string spacetime [44], in the spacetime with torsion [42,45], under noninertial effects [46,47], in two-dimensional scenarios [48,49], in the cosmic string spacetime [50], and in possible scenarios of Lorentz symmetry violation [51]. Then, by substituting Equation (10) and the non-minimal coupling of the DO into Equation (11), we have

$$\begin{array}{c}\hfill \left[i{\gamma }^{\left(t\right)}\left({\partial }_t-i{\displaystyle \frac{Z{e}^2}{r}}\right)+i{\gamma }^{\left(r\right)}\left(\alpha {\partial }_r+{\displaystyle \frac{(\alpha -1)}{r}}+m\omega \alpha \widehat{\beta }r\right)+i{\displaystyle \frac{{\gamma }^{\left(\theta \right)}}{r}}{\partial }_{\theta }+i{\displaystyle \frac{{\gamma }^{\left(\phi \right)}}{rsin\theta }}{\partial }_{\phi }-m\right]\psi =0.\end{array}$$

(13)

Equation (13) describes the HA under the effects of the DO in GM spacetime. In order to obtain a purely analytical solution to this problem, let us consider the set of solutions to the spinor wave equation [2,52]:

$$\begin{array}{c}\hfill \psi (\overrightarrow{r},t)={\displaystyle \frac{1}{r}}\left[\begin{array}{c}if\left(r\right){\Phi }_{j,m}(\theta ,\phi )\\ g\left(r\right)(\overrightarrow{\sigma }.\widehat{r}){\Phi }_{j,m}(\theta ,\phi )\end{array}\right]e^{-i\mathcal{E}t},\end{array}$$

(14)

in which ${\Phi }_{j,m}$ are the spinor spherical harmonics, $f\left(r\right)$ and $g\left(r\right)$ are radial wave spinor functions, and $\mathcal{E}$ is the parameter associated to the energy of the system. Note that Equation (14) presents well-defined parity under the transformation $\overrightarrow{r}\to \overrightarrow{r^{\prime }}=-\overrightarrow{r}$ [52]. Hence, by substituting Equation (14) into Equation (13), we obtain radial differential equations

$$\begin{array}{ccc}\hfill \left(\mathcal{E}-m-{\displaystyle \frac{Z{e}^2}{r}}\right)f+\alpha {\displaystyle \frac{dg}{dr}}-{\displaystyle \frac{\kappa }{r}}g-m\omega \alpha rg& =& 0;\hfill \\ \hfill \left(\mathcal{E}+m-{\displaystyle \frac{Z{e}^2}{r}}\right)g-\alpha {\displaystyle \frac{df}{dr}}-{\displaystyle \frac{\kappa }{r}}f-m\omega \alpha rf& =& 0,\hfill \end{array}$$

(15)

for which we use the definition [2,52]

$$\begin{array}{c}\hfill (\overrightarrow{\sigma }.\overrightarrow{A})(\overrightarrow{\sigma }.\overrightarrow{B})=\overrightarrow{A}.\overrightarrow{B}+i\overrightarrow{\sigma }.(\overrightarrow{A}\times \overrightarrow{B});\overrightarrow{\sigma }.\overrightarrow{L}{\Phi }_{j,m}=-(1+\kappa ){\Phi }_{j,m};\overrightarrow{L}=-i\left(\widehat{\phi }{\displaystyle \frac{\partial }{\partial \theta }}-{\displaystyle \frac{\widehat{\theta }}{sin\theta }}{\displaystyle \frac{\partial }{\partial \phi }}\right),\end{array}$$

(16)

with $\kappa =\mp j+{\displaystyle \frac{1}{2}}$, $j=l\pm {\displaystyle \frac{1}{2}}$, and $l=0,1,2,\dots$

By multiplying the first equation of Equation (15) by $\left(E+m-{\displaystyle \frac{Z{e}^2}{r}}\right)$ and by using the second equation of Equation (15), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}f}{d{r}^2}}-{\displaystyle \frac{A^2}{r^2}}f-{\displaystyle \frac{B^2}{r}}f-C^2{r}^{2}f+D^{2}f=0,\end{array}$$

(17)

where we define the new parameters

$$\begin{array}{c}\hfill A^2={\displaystyle \frac{\kappa (\kappa +\alpha )-{\left(Z{e}^2\right)}^2}{{\alpha }^2}};B^2={\displaystyle \frac{2Z{e}^2\mathcal{E}}{{\alpha }^2}};C^2=m^2{\omega }^2;D^2={\displaystyle \frac{{\mathcal{E}}^2-m^2+m\omega \alpha (\alpha -\kappa )}{{\alpha }^2}}.\end{array}$$

(18)

Now, let us consider the change of variable $s\sqrt{C}r$ into Equation (17), from which we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}f}{d{s}^2}}-{\displaystyle \frac{A^2}{s^2}}f-{\displaystyle \frac{B}{\sqrt{C}}}{\displaystyle \frac{f}{s}}-s^{2}f+{\displaystyle \frac{D^2}{C}}f=0\end{array}$$

(19)

By analyzing the asymptotic behavior of Equation (19), it is possible to propose a general solution as follows:

$$\begin{array}{c}\hfill f\left(s\right)=s^{{\displaystyle \frac{\sqrt{4A^2+1}+1}{2}}}{e}^{-{\displaystyle \frac{s^2}{2}}}F\left(s\right),\end{array}$$

(20)

where $F\left(s\right)$ is currently an unknown function. Therefore, by substituting Equation (20) into Equation (19), we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}F}{d{s}^2}}+\left[{\displaystyle \frac{\sqrt{4A^2+1}+1}{s}}-2s\right]{\displaystyle \frac{dF}{ds}}+\left[{\displaystyle \frac{D^2}{C}}-(\sqrt{4A^2+1}+2)-{\displaystyle \frac{B^2}{\sqrt{C}s}}\right]F=0.\end{array}$$

(21)

Equation (21) is known in the literature as a biconfluent Heun equation [32,53], and its solution is the biconfluent Heun function [53]:

$$\begin{array}{c}\hfill F\left(s\right)=h_b\left(\sqrt{4A^2+1},0,{\displaystyle \frac{D^2}{C}},{\displaystyle \frac{2B^2}{\sqrt{C}}};s\right)\end{array}$$

(22)

The biconfluent heun equation has two singular points, the origin and infinity, where the first is a regular singular point, while the second is an irregular singular point. As a result, Equation (21) has at least one solution around the origin, given in the form of a power series [54]:

$$\begin{array}{c}\hfill F\left(s\right)=\sum _{j=0}^{\infty }{a}_j{s}^j.\end{array}$$

(23)

Therefore, by substituting Equation (23) into Equation (22), we obtain the recurrence relation for the coefficients of power series

$$\begin{array}{c}\hfill a_{j+2}={\displaystyle \frac{(B^2/\sqrt{C})a_{j+1}-[D^2/C-(\sqrt{4A^2+1}+2+2j)]a_j}{(j+2)(j+2+\sqrt{4A^2+1})}},\end{array}$$

(24)

with

$$\begin{array}{ccc}\hfill a_1& =& {\displaystyle \frac{B^2}{(1+\sqrt{4A^2+1})\sqrt{C}}}{a}_0;\hfill \\ \hfill a_2& =& {\displaystyle \frac{a_0}{2C(2+\sqrt{4A^2+1})}}\left\{{\displaystyle \frac{B^4}{(1+\sqrt{4A^2+1})}}+C(\sqrt{4A^2+1}+2)-D^2\right\}.\hfill \end{array}$$

(25)

As we are interested in solutions of bound state, we must truncate the power series in order to obtain polynomials of degree n. Therefore, from the recurrence relation given in Equation (24), we obtain two conditions to truncate the biconfluent Heun series:

$$\begin{array}{c}\hfill a_{n+1}=0;{\displaystyle \frac{D^2}{C}}-\sqrt{4A^2+1}-2=2n\end{array}$$

(26)

with $n=1,2,3,\dots$

We can note that we must impose values for n in the series truncation conditions to obtain any physical information. In this sense, let us consider $n=1$, which, from a physical point of view, represents the ground state of the system. Then, for $n=1$ in Equation (26), we have

$$\begin{array}{c}\hfill a_2=0;D^2=C(4+\sqrt{4A^2+1}).\end{array}$$

(27)

From the condition $D^2=C(4+\sqrt{4A^2+1})$, we obtain

$$\begin{array}{c}\hfill {\mathcal{E}}_{l,1}^2=m^2+m\omega {\alpha }^2\left(3+{\displaystyle \frac{\kappa +\sqrt{4\kappa (\kappa +\alpha )-{\left(2Z{e}^2\right)}^2+\alpha }}{\alpha }}\right).\end{array}$$

(28)

Our analysis is not complete; we need to consider the condition $a_2=0$. For this, let us to adjust the angular frequency of the DO imposed so that $a_2=0$ and $a_{n+1}=0$ for any value of the radial modes n. Therefore, by isolating $\omega$ from $a_2=0$, we have the following:

$$\begin{array}{c}\hfill {\omega }_{l,1}={\displaystyle \frac{[(1+\sqrt{4A^2+1}){\alpha }^2-{\left(2Z{e}^2\right)}^2]}{(1+\sqrt{4A^2+1}+\kappa )(1+\sqrt{4A^2+1})}}{\displaystyle \frac{{\mathcal{E}}_{l,1}^2}{m{\alpha }^4}}-{\displaystyle \frac{m}{\alpha [\alpha (1+\sqrt{4A^2+1})+\kappa ]}},\end{array}$$

(29)

where we have to label the angular frequency of the DO due its dependency on the quantum numbers of the system, that is, $\omega ={\omega }_{l,n}$. We can note that, in contrast to Refs. [26,38,39], the angular frequency of the DO has allowed for values that depend on the quantum numbers $\{l,n\}$. It is important to observe that this quantum effect is produced by the interaction between Coulomb potential and the DO, that is, the angular frequency of DO depends on the quantum number of the system due the presence of Coulomb potential. In addition, we can see that the allowed values of the angular frequency of DO are influenced by the topological defect. This influence is explicit due to the presence of the parameter $\alpha$, associated with GM in Equation (29).

Due to this quantum effect on the angular frequency of the DO, Equation (28) can be rewritten as follows:

$$\begin{array}{c}\hfill {\mathcal{E}}_{l,1}^2=m^2+m{\omega }_{l,1}{\alpha }^2\left(3+{\displaystyle \frac{\kappa +\sqrt{4\kappa (\kappa +\alpha )-{\left(2Z{e}^2\right)}^2+\alpha }}{\alpha }}\right).\end{array}$$

(30)

Then, for our analysis to be complete, let us to substitute Equation (29) into Equation (30), where we obtain

$$\begin{array}{c}\hfill {\mathcal{E}}_{l,1}=\pm m{\left\{{\displaystyle \frac{1-\frac{\alpha }{[\alpha (1+\sqrt{4A^2+1})+\kappa ]}}{\left[1-\frac{[(1+\sqrt{4A^2+1}){\alpha }^2-{\left(2Z{e}^2\right)}^2]}{{\alpha }^2(1+\sqrt{4A^2+1}+\kappa )(1+\sqrt{4A^2+1})}\right]}}\right\}}^{1/2}.\end{array}$$

(31)

Equation (31) represents the allowed values of the ground state of the HA under the effects of the DO in the GM spacetime. We can observe that the relativistic energy profile of system is drastically modified. This modification can be analyzed from two perspectives. The first concerns considering the energy profile of the HA as a reference. In this sense, by comparing the result given in Equation (31) with Refs. [2,25,27], we can note that the presence of the DO modifies the relativistic energy profile of the system. This change is noted through the expression given in Equation (31), which is not a closed expression for any radial mode, but rather an expression corresponding only to the allowed values of the ground state defined by the radial mode $n=1$. The other perspective is to consider the DO as a reference. Therefore, comparing Equation (31) with the results provided in Refs. [26,40], we can note that the presence of Coulomb potential in the system modifies the energy profile of the DO, that is, contrary to Refs. [26,40], it is not possible to obtain a unique expression to describe the relativistic energy levels for any radial mode n, but rather an expression of allowed values for each radial mode n. In addition, we can also investigate the influence of the topological defect on the allowed values of relativistic energy for the ground state. By taking $\alpha \to 1$, we recover the allowed values of relativistic energy for the ground state of the HA plus the DO.

3. PA Under Effects of KGO in GM Spacetime

From now on, let us describe the PA under the effects of oscillations described by the KGO in the spacetime of the GM, defined by the metric given in Equation (1). To do this, we must consider the Klein–Gordon equation with minimal coupling for KGO given in the following form [55,56]:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\sqrt{-g}}}({\partial }_{\mu }+m\omega X_{\mu }-ie{A}_{\mu })\left(\sqrt{-g}{g}^{\mu \nu }\right)({\partial }_{\nu }-m\omega X_{\nu }-ie{A}_{\nu })\varphi -\xi R\varphi -m^2\varphi =0,\end{array}$$

(32)

where $\xi$ is an arbitrary coupling constant, while $g=\det \left(g_{\mu \nu }\right)$ and $g^{\mu \nu }$ is an inverse metric tensor. In this way, from Equations (1) and (10), Equation (32) becomes

$$\begin{array}{ccc}\hfill -{\left({\partial }_t+ie{A}_0\right)}^2\varphi & +& {\displaystyle \frac{{\alpha }^2}{r^2}}({\partial }_r+m\omega r)(r^2{\partial }_r\varphi -m\omega r^3\varphi )+{\displaystyle \frac{1}{r^{2}sin\theta }}{\partial }_{\theta }(sin\theta {\partial }_{\theta }\varphi )+{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\partial }_{\phi }^2\varphi \hfill \\ & -& {\displaystyle \frac{2\xi (1-{\alpha }^2)}{r^2}}\varphi -m^2\varphi =0.\hfill \end{array}$$

(33)

In order to describe the quantum dynamics of PA subject to KGO, let us consider the general solution given in the following form:

$$\begin{array}{c}\hfill \varphi (r,\theta ,\phi ,t)=e^{-i\mathcal{E}t}R\left(r\right)Y_{l,m}(\theta ,\phi ),\end{array}$$

(34)

where $Y_{l,m}(\theta ,\phi )$ represents the well-known spherical harmonics, and $R\left(r\right)$ is a radial wave function. In this way, by considering the definition

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{1}{sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right)+{\displaystyle \frac{1}{{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\phi }^2}}\right]Y_{l,m}(\theta ,\phi )=-l(l+1)Y_{l,m}(\theta ,\phi )\end{array}$$

(35)

and considering that $A_0=-Ze/r$, by substituting Equation (34) into Equation (33), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}R}{d{r}^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{dR}{dr}}-{\displaystyle \frac{a^2}{r^2}}R-{\displaystyle \frac{B}{r}}R-C^2{r}^{2}R+d^{2}R=0,\end{array}$$

(36)

where we define the news parameters

$$\begin{array}{c}\hfill a^2={\displaystyle \frac{l(l+1)+2\xi (1-{\alpha }^2)-{\left(Z{e}^2\right)}^2}{{\alpha }^2}};d^2={\displaystyle \frac{{\mathcal{E}}^2-m^2-3m\omega {\alpha }^2}{{\alpha }^2}},\end{array}$$

(37)

with B and C defined in Equation (18).

Now, let us integrate the variable change $s=\sqrt{C}r$ into Equation (36), from which we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}R}{d{s}^2}}+{\displaystyle \frac{2}{s}}{\displaystyle \frac{dR}{ds}}-{\displaystyle \frac{a^2}{s^2}}R-{\displaystyle \frac{B^2}{\sqrt{C}s}}R-s^{2}R+{\displaystyle \frac{d^2}{C}}R=0.\end{array}$$

(38)

The general solution of Equation (38) is given in form

$$\begin{array}{c}\hfill R\left(s\right)=s^{{\displaystyle \frac{\sqrt{4a^2+1}}{2}}-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{s^2}{2}}}g\left(s\right).\end{array}$$

(39)

Then, by substituting Equation (39) into Equation (38), we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}g}{d{s}^2}}+\left[{\displaystyle \frac{\sqrt{4a^2+1}+1}{s}}-2s\right]{\displaystyle \frac{dg}{ds}}+\left[{\displaystyle \frac{d^2}{C}}-(\sqrt{4a^2+1}+2)-{\displaystyle \frac{B^2}{\sqrt{C}s}}\right]g=0,\end{array}$$

(40)

which is analogous to Equation (21), that is, the biconfluent Heun equation [32,53], and $g\left(s\right)$ is the biconfluent Heun function [53]:

$$\begin{array}{c}\hfill g\left(s\right)=H_b\left(\sqrt{4a^2+1},0,{\displaystyle \frac{d^2}{C}},{\displaystyle \frac{2B^2}{\sqrt{C}}};s\right).\end{array}$$

(41)

Therefore, following the same steps given in Equation (23) to Equation (31), we obtain the expressions

$$\begin{array}{c}\hfill {\omega }_{l,1}={\displaystyle \frac{{\mathcal{E}}_{l,1}^2}{{\alpha }^{2}m(5+\sqrt{4a^2+1})}}\left[1-{\displaystyle \frac{4{\left(Z{e}^2\right)}^2}{{\alpha }^2(1+\sqrt{4a^2+1})}}\right]-{\displaystyle \frac{m}{{\alpha }^2(5+\sqrt{4a^2+1})}}\end{array}$$

(42)

and

$$\begin{array}{c}\hfill {\mathcal{E}}_{l,1}=\pm m\sqrt{{\displaystyle \frac{1}{1-\frac{{\left(2Z{e}^2\right)}^2}{{\alpha }^2(1+\sqrt{4a^2+1})}}}},\end{array}$$

(43)

which represent the allowed values of angular frequency and relativistic energy of the ground state of the system analyzed, respectively. Again, we can analyze the above result from two perspectives. By comparing the result given in Equation (43) with the result given in Ref. [25], we can note that the presence of KGO on PA modifies the relativistic energy levels. This modification is explicit through the corresponding expression, that is, Equation (43) is not a unique expression that describes the relativistic energy profile for any value of radial modes n; Equation (43) represents the lowest energy values of the system characterized by the $n=1$ radial mode. On the other hand, by comparing Equation (42) with Refs. [26,40], we can observe that the presence of Coulomb potential on KGO modifies the energy levels of the relativistic oscillator model for spin-0 particles, that is, in addition to it not being possible to determine a unique expression for the relativistic energy profile of KGO for any value of n, its fundamental state is no longer defined by $n=0$, but by $n=1$. We can note also the influence of the topological defect on the allowed values of relativistic energy for the ground state. By taking $\alpha \to 1$, we recover the allowed values of relativistic energy for the ground state of PA plus KGO.

4. Conclusions

We investigated HAs and PAs under the effects of oscillations characterized by the presence of DO and KGO, respectively, in the GM spacetime. First, we analyzed HAs interacting with DO and then followed the same process for PAs subject to KGO. Through a purely analytical analysis, we determined the relativistic energy profile of the system for two cases. In these two cases, we can observe that the relativistic energy profile of the system is drastically modified. The modification that draws the most attention is the impossibility of describing the energy levels of the proposed systems with a single mathematical expression for any radial mode. It is only possible to determine expressions for allowed values for radial modes separately, as found for the ground state presented in Equations (31) and (43). From the point of view of relativistic oscillator models, we can note that the ground state is no longer defined by $n=0$ [26,38,40], but by $n=1$, a quantum effect arising from the presence of the Coulomb interaction.

Another quantum effect is produced by the presence of Coulomb potential, which is the dependence of the angular frequencies of the DO and KGO on the quantum numbers associated with the angular momentum and radial modes, that is, differing from Refs. [26,38,40], the angular frequencies of relativistic oscillator models have well-defined restricted values, which implies that, in both cases, the angular frequency is quantized.

In both analyzed cases, the lowest allowed values of the system energy are influenced by the topological defect present in spacetime. Hence, by making $\alpha \to 1$, we recover the lowest allowed values of the system energy of the HA and PA under the effects of DO and KGO, respectively, in the Minkowski spacetime.