1. Introduction
Currently, the relativistic quantum dynamics of spinless particles in the presence of external fields is of great interest to researchers. The physical properties of these systems are accessed by the solution of the Klein–Gordon field equation with electromagnetic interactions [1, 2]. The electromagnetic interactions are introduced into the Klein–Gordon equation through the so called minimal substitution, ${p}_{\mu }\to {p}_{\mu }-e\,{A}_{\mu }$ , where e is the charge and Aμ is the four-vector potential of the electromagnetic field.
In recent years, there has been growing interest in finding the solution of relativistic wave equations of spin-zero particles, with or without potential, of various kinds, and using a variety of methods. In particular, several researchers have investigated the physical properties of a series of backgrounds with Gödel-type geometries. The relativistic quantum dynamics of spin-zero scalar particles in the Som–Raychaudhuri geometries was investigated in [3, 4] and observed the similarity of the energy eigenvalue with the Landau levels [5] in flat spaces (see also, [6]). The relativistic quantum dynamics of a scalar particle in the Gödel-type geometries of flat, spherical, and hyperbolic spaces was investigated in [7]. The Klein–Gordon oscillator under the influence of topological defects in the presence of external fields in the Som–Raychaudhuri space–time was studied in [8]. The relativistic quantum dynamics of a scalar particle in a Gödel-type metric with cosmic string was investigated in [9] and compared its results with landau levels in flat spaces. A quantum particle described by the Klein–Gordon oscillator interacting with a cosmic dislocation in Som–Raychaudhuri space–time, and in the presence of a ahomogeneous magnetic field using the Kaluza–Klein theory was investigated in [10]. Dirac fermions in a Gödel-type background space–times with torsion were investigated in [11]. Weyl fermions in a family of Gödel-type geometries as per Einstein’s theory of general relativity was studied in [12]. The authors solved the Weyl equation and found the energy eigenvalue and eigen-spinors for all three cases of Gödel-type geometries where the topological defecta is passing through string. In [13], a photon equation (mass-less Duffin–Kemmer–Petiau equation) was written explicitly for the general type of stationary Gödel space–time, and was solved exactly both for Gödel-type and Gödel space–times. The relativistic quantum dynamics of a scalar particle in a topologically trivial flat Gödel-type space–time was studied in [14]. Linear confinement of a scalar particle in the Som–Raychaudhuri space–time with a cosmic string in [15] (see also, [16]) and with a scalar and vector potentials of Coulomb-type in a topologically trivial flat Gödel-type space–time, were investigated in [17]. The Dirac equation in a topologically trivial flat Gödel-type space–time was investigated in [18]. Spin-0 system of the DKP equation and the DKP oscillator in a topologically trivial flat Gödel-type space–time was investigated in [19] and [20], respectively. The generalized Klein–Gordon oscillator subject to a scalar Coulomb potential in the backgrounds of (1+2)-dimensional Gürses space–time was investigated in [21]. The behavior of scalar particles with Yukawa-like confining potential in the Som–Raychaudhuri space–time in the presence of topological defects was investigated in [22]. The ground state of a bosonic massive charged particle in the presence of external fields in the Som–Raychaudhuri space–time was investigated in [23]. Recently, we pointed out in [24] that the derived radial wave equation of the Klein–Gordon equation in the Som–Raychaudhuri space–time in the presence of external fields in [23] was incorrect. The relativistic quantum dynamics of spin-0 massive charged particles in the presence of external fields in a four-dimensional curved space–time was investigated in [25]. The Klein–Gordon equation with vector and scalar potentials of a Coulomb-type under the influence of non-inertial effects in the cosmic string space–time was studied in [26]. The confinement of scalar particles subject to Coulomb-type potential has been investigated by many authors (e.g., [27–30]). It is also worth mentioning studies that have dealt with Coulomb-type potential in the propagation of gravitational waves [31], quark models [32] and in relativistic quantum mechanics [33–36].
In this paper, we investigate the analog effect on the Aharonov–Bohm effect for bound states in spin-0 massive charged particles in a Gödel-type space–time in the presence of external fields. We extend this analysis, including a Coulomb potential. We derive and solve the radial wave equation of the Klein–Gordon equation in the system under consideration, and obtain relativistic energy eigenvalue and eigenfunctions. The energy eigenvalue obtained here reduces to the results in [3, 9] mentioned in other research.
2. Bosonic charged particles: the Klein–Gordon equation
The relativistic quantum dynamics of a spin-zero massive charged particle of mass M is described by the following KG equation [36]: $ \begin{eqnarray}\left[\displaystyle \frac{1}{\sqrt{-g}}\,{{\mathsf{D}}}_{\mu }(\sqrt{-g}\,{g}^{\mu \nu }\,{{\mathsf{D}}}_{\nu })-\xi \,R\right]\,{\rm{\Psi }}={M}^{2}\,{\rm{\Psi }},\end{eqnarray}$ where ${{\mathsf{D}}}_{\mu }\equiv {{\rm{\partial }}}_{\mu }-{\mathsf{i}}\,e\,{A}_{\mu }$ is the minimal coupling, ${\partial }_{\mu }=\tfrac{\partial }{\partial \,{x}^{\mu }},e$ is the electric charge, Aμ is the electromagnetic four-vector potential, ξ is the non-minimal coupling constant with background curvature, and R is the scalar curvature.
We have selected the four-vector electromagnetic potential ${A}_{\mu }=(-{A}_{0},\vec{A})$ with $ \begin{eqnarray}{A}_{0}=V,\quad \vec{A}=(0,{A}_{\phi },0),\end{eqnarray}$ where the temporal component is represented by a scalar potential V.
Consider the Som–Raychaudhuri space–time with the cosmic string given by [8, 9]: $ \begin{eqnarray}{\rm{d}}{s}^{{\mathsf{2}}}=-{\left({\rm{d}}t+\alpha {\rm{\Omega }}{r}^{{\mathsf{2}}}{\rm{d}}\phi \right)}^{{\mathsf{2}}}+{\alpha }^{{\mathsf{2}}}\,{r}^{{\mathsf{2}}}\,{\rm{d}}{\phi }^{{\mathsf{2}}}+{\rm{d}}{r}^{{\mathsf{2}}}+{\rm{d}}{z}^{{\mathsf{2}}},\end{eqnarray}$ where α and Ω respectively characterize the cosmic string and vorticity parameter of space–time. The cylindrical coordinates are in the ranges $-\infty \lt (t,z)\lt \infty ,0\leqslant r\lt \infty $ , and the coordinate φ is periodic. Note that the above metric is of $(-,\,+,\,+,\,+)$ or +2 signature, and is Lorentzian, with ${\mathsf{\det }}\,g=-{\alpha }^{2}\,{r}^{2}$ . The scalar curvature of the above metric is R = 2 Ω2.
For geometry (3 ), the KG equation (1 ) becomes $ \begin{eqnarray}\begin{array}{l}\left[-{\left(\displaystyle \frac{\partial }{\partial t}+{\mathsf{i}}e{A}_{0}\right)}^{2}+\left\{\displaystyle \frac{1}{\alpha \,r}\left(\displaystyle \frac{\partial }{\partial \phi }-{\mathsf{i}}\,e\,{A}_{\phi }\right)\right.\right.\\ \quad {\left.-{\rm{\Omega }}r\left(\displaystyle \frac{\partial }{\partial t}+{\mathsf{i}}e{A}_{0}\right)\right\}}^{2}+\displaystyle \frac{1}{r}\,\displaystyle \frac{\partial }{\partial r}\left(r\,\displaystyle \frac{\partial }{\partial r}\right)\\ \quad \left.+\,{\partial }_{z}^{2}-({M}^{2}+2\,\xi \,{{\rm{\Omega }}}^{2})\right]\,{\rm{\Psi }}=0.\end{array}\end{eqnarray}$
The above equation is independent of time, and symmetrical by translations along the z-axis as well by rotation. It is reasonable to write the solution to equation (4 ) as $ \begin{eqnarray}{\rm{\Psi }}(t,r,\phi ,z)={{\rm{e}}}^{{\mathsf{i}}(-Et+l\phi +kz)}\,\psi (r),\end{eqnarray}$ where E is the total energy, l = 0, ±1, ±2, …, are the eigenvalues of the z-component of the angular momentum operator, and k represents the eigenvalues of the z-component of the linear momentum operator.
Substituting solution (5 ) into equation (4 ), we obtain the following equation for the radial wave-function ψ (r): $ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{\psi }^{{\prime\prime} }(r)+\displaystyle \frac{1}{r}\,{\psi }^{{\prime} }(r)+\left[{\left(E-{eV}\right)}^{2}-{M}^{2}-{k}^{2}\right.\\ \,\,-\,2\,\xi \,{{\rm{\Omega }}}^{2}-\displaystyle \frac{{\left(l-e{A}_{\phi }\right)}^{2}}{{\alpha }^{2}\,{r}^{2}}\\ \,\,-\,{{\rm{\Omega }}}^{2}\,{r}^{2}{\left(E-{eV}\right)}^{2}\\ \,\,\left.-\displaystyle \frac{2\,{\rm{\Omega }}(E-e\,V)}{\alpha }(l-e\,{A}_{\phi })\right]\,\psi (r)=0.\end{array}\end{array}\end{eqnarray}$
We discuss two cases of the above equation as follows:
Case 1: considering the scalar potential V = 0 in the above relativistic system.
Let us assume that the topological defects (cosmic string) have an internal magnetic flux field (with magnetic flux ΦB) [37]. Thereby, we obtain the angular component of the electromagnetic four-vector potential given by [30, 38–40] $ \begin{eqnarray}e\,{A}_{\phi }={\rm{\Phi }},\quad {\rm{\Phi }}=\displaystyle \frac{{{\rm{\Phi }}}_{B}}{(2\,\pi /e)},\end{eqnarray}$ where ΦB is the magnetic quantum flux.
Substituting the above electromagnetic potential equation (7 ) into equation (6 ), we obtain the following differential equation: $ \begin{eqnarray}\psi ^{\prime\prime} (r)+\displaystyle \frac{1}{r}\,\psi ^{\prime} (r)+\left[\lambda -{{\rm{\Omega }}}^{2}\,{E}^{2}\,{r}^{2}-\displaystyle \frac{{l}_{0}^{2}}{{r}^{2}}\right]\,\psi (r)=0,\end{eqnarray}$ where $ \begin{eqnarray}\lambda ={E}^{2}-{M}^{2}-{k}^{2}-2\,\xi \,{{\rm{\Omega }}}^{2}-2\,{\rm{\Omega }}\,E\,{l}_{0},\quad {l}_{0}=\displaystyle \frac{(l-{\rm{\Phi }})}{\alpha }.\end{eqnarray}$ Transforming $x={\rm{\Omega }}E{r}^{2}$ into the above equation (8 ), we obtain the following differential equation [41]: $ \begin{eqnarray}\psi ^{\prime\prime} (x)+\displaystyle \frac{1}{x}\,\psi ^{\prime} (x)+\displaystyle \frac{1}{{x}^{2}}(-{\xi }_{1}\,{x}^{2}+{\xi }_{2}\,x-{\xi }_{3})\psi (x)=0,\end{eqnarray}$ where $ \begin{eqnarray}{\xi }_{1}=\displaystyle \frac{1}{4},\quad {\xi }_{2}=\displaystyle \frac{\lambda }{4\,{\rm{\Omega }}\,E},\quad {\xi }_{3}=\displaystyle \frac{{l}_{0}^{2}}{4}.\end{eqnarray}$ The second degree energy eigenvalues equation is given by $ \begin{eqnarray}\begin{array}{l}{E}_{n,l}^{2}-2\,{\rm{\Omega }}\,{E}_{n,l}(2\,n+1+| {l}_{0}| +{l}_{0})\\ \quad -\,{M}^{2}-{k}^{2}-2\,\xi \,{{\rm{\Omega }}}^{2}=0.\end{array}\end{eqnarray}$ The energy eigenvalues associated with nth radial modes is $ \begin{eqnarray}\begin{array}{l}{E}_{n,l}={\rm{\Omega }}\left(2\,n+1+\displaystyle \frac{(l-{\rm{\Phi }})}{\alpha }+\displaystyle \frac{| l-{\rm{\Phi }}| }{\alpha }\right)\\ \pm \,\sqrt{{{\rm{\Omega }}}^{2}{\left(2n+1+\displaystyle \frac{(l-{\rm{\Phi }})}{\alpha }+\displaystyle \frac{| l-{\rm{\Phi }}| }{\alpha }\right)}^{2}+{M}^{2}+{k}^{2}+2\,\xi \,{{\rm{\Omega }}}^{2}},\end{array}\end{eqnarray}$ where n = 0, 1, 2, 3, ..., .
Equation (13 ) is the relativistic energy eigenvalue of a massive charged particle in the presence of an internal magnetic flux field with background curvature in a Gödel-type space–time. The energy eigenvalues equation (13 ) depends on physical parameters such as the cosmic string α, the Aharonov–Bohm magnetic flux ΦB, and the non-minimal coupling constant ξ. The relativistic energy eigenvalue equation (13 ) depends on the geometric quantum phase [37]. This dependence of the relativistic energy eigenvalue on the geometric quantum phase gives rise to an analogous effect on the Aharonov–Bohm effect for bound states [37, 42–44]. Thus, for ${{\rm{\Phi }}}_{0}=\pm \tfrac{2\,\pi \,\alpha }{e}\,\tau $ , we find that ${E}_{n,\bar{l}}({{\rm{\Phi }}}_{B}+{{\rm{\Phi }}}_{0})\,={E}_{n,\bar{l}\mp \tau }({{\rm{\Phi }}}_{B})$ where, $\bar{l}=\tfrac{l}{\alpha }$ and τ = 1, 2, 3, ..., .
We can see in comparison to the results without electromagnetic interactions as obtained in [3, 9], that the energy eigenvalues equation (13 ) of a massive charged particle are modified, even though there is no direct interaction of the particle with the external magnetic fields. For ${{\rm{\Phi }}}_{B}\to 0$ , and without non-minimal coupling $\xi \to 0$ , the energy eigenvalue equation (13 ) reduces to the result obtained in [9]. Again, for ${{\rm{\Phi }}}_{B}\to 0,\xi \to 0$ , and $\alpha \to 1$ , the energy eigenvalues equation (13 ) reduces to the result obtained in [3].
The normalized eigenfunction is given by $ \begin{eqnarray}{\psi }_{n,l}(x)=| N{| }_{n}\,{x}^{\tfrac{| l-{\rm{\Phi }}| }{2\alpha }}\,{{\rm{e}}}^{-\tfrac{x}{2}}\,{{\mathsf{L}}}_{n}^{(\tfrac{| l-{\rm{\Phi }}| }{\alpha })}(x),\end{eqnarray}$ where $| N{| }_{n}={\left(\displaystyle \frac{n!}{\left(n+\displaystyle \frac{| l-{\rm{\Phi }}| }{\alpha }\right)!}\right)}^{\displaystyle \frac{1}{2}}$
is the normalization constant and${{\mathsf{L}}}_{n}^{(\kappa )}(x)$ gives the generalized Laguerre polynomials, which are orthogonal over $[0,\infty )$ with respect to the measure with weighting function ${x}^{\kappa }\,{{\rm{e}}}^{-x}$ as $ \begin{eqnarray}{\int }_{0}^{\infty }{x}^{\kappa }\,{{\rm{e}}}^{-x}\,{{\mathsf{L}}}_{n}^{(\kappa )}\,{{\mathsf{L}}}_{m}^{(\kappa )}\,{\rm{d}}x=\displaystyle \frac{(n+\kappa )!}{n!}\,{\delta }_{nm}.\end{eqnarray}$
is the normalization constant and
Case 2: considering the scalar potential to be Coulomb-type $V=\tfrac{{\xi }_{c}}{r}$ where, ξc > 0.
Substituting the above potential with (7 ) into equation (6 ), we obtain the following radial wave equation for ψ (r): $ \begin{eqnarray}\begin{array}{l}\psi ^{\prime\prime} (r)+\displaystyle \frac{1}{r}\,\psi ^{\prime} (r)\\ \quad +\,\left[{\lambda }_{0}-{\omega }^{2}\,{r}^{2}-\displaystyle \frac{{j}^{2}}{{r}^{2}}-\displaystyle \frac{a}{r}+b\,r\right]\,\psi (r)=0,\end{array}\end{eqnarray}$ where $ \begin{eqnarray}\begin{array}{rcl}{\lambda }_{0} & = & {E}^{2}-{M}^{2}-{k}^{2}-{{\rm{\Omega }}}^{2}\,{e}^{2}\,{\xi }_{c}^{2}-2\,{\rm{\Omega }}\,E\,{l}_{0}-2\,\xi \,{{\rm{\Omega }}}^{2},\\ \omega & = & {\rm{\Omega }}\,E,\\ j & = & \sqrt{{l}_{0}^{2}-{e}^{2}\,{\xi }_{c}^{2}},\\ a & = & 2\,e\,{\xi }_{c}(E-{\rm{\Omega }}\,{l}_{0}),\\ b & = & 2\,{{\rm{\Omega }}}^{2}\,E\,e\,{\xi }_{c}.\end{array}\end{eqnarray}$ Transforming $x=\sqrt{\omega }\,r$ into the above equation (16 ), we obtain the following wave equation: $ \begin{eqnarray}\psi ^{\prime\prime} (x)+\displaystyle \frac{1}{x}\,\psi ^{\prime} (x)+\left[\zeta -{x}^{2}-\displaystyle \frac{{j}^{2}}{{x}^{2}}-\displaystyle \frac{\eta }{x}+\theta \,x\right]\,\psi (x)=0,\end{eqnarray}$ where $ \begin{eqnarray}\zeta =\displaystyle \frac{{\lambda }_{0}}{\omega },\quad \eta =\displaystyle \frac{a}{\sqrt{\omega }},\quad \theta =\displaystyle \frac{b}{{\omega }^{\tfrac{3}{2}}}.\end{eqnarray}$
Next, we use the appropriate boundary conditions to investigate the bound states solution in this problem. It is a requirement that the wave-functions must be regular both at $x\to 0$ and $x\to \infty $ . We then proceed with an analysis of the asymptotic behavior of the radial eigenfunction at origin and in the infinite. These conditions are necessary, since the wave-functions must be well-behaved within these limits, thereby enabling us to obtain the bound states of the energy for the system. Let us impose the requirement that $\psi (x)\to 0$ when $x\to 0$ and $x\to \infty $ . Suppose the possible solution to the equation (18 ) is $ \begin{eqnarray}\psi (x)={x}^{A}\,{{\rm{e}}}^{-(Bx+D{x}^{2})}\,H(x),\end{eqnarray}$ where H(x) is an unknown function. Substituting solution (20 ) into equation (18 ), we obtain $ \begin{eqnarray}\begin{array}{l}H^{\prime\prime} (x)+\left(\displaystyle \frac{1+2\,A}{x}-2\,B-4\,D\,x\right)H^{\prime} (x)\\ \quad +\,\left[\displaystyle \frac{{A}^{2}-{j}^{2}}{{x}^{2}}-\displaystyle \frac{\eta +(1+2\,A)B}{x}\right.\\ \quad +(\zeta +{B}^{2}-4\,D(1+A))+(4\,B\,D+\theta )x\\ \quad +\left.\,(4\,{D}^{2}-1){x}^{2}\right]\,H(x)=0.\end{array}\end{eqnarray}$ Equating the coefficients of x−2, x1, x2 to zero in the above equation gives $ \begin{eqnarray}\begin{array}{rcl}{x}^{-2} & : & \quad {A}^{2}-{j}^{2}=0\Rightarrow A=j,\\ {x}^{1} & : & \quad 4\,B\,D+\theta =0\Rightarrow B=-\displaystyle \frac{\theta }{2},\\ {x}^{2} & : & \quad 4\,{D}^{2}-1=0\Rightarrow D=\displaystyle \frac{1}{2}.\end{array}\end{eqnarray}$ With these, the above equation (21 ) can now be expressed as $ \begin{eqnarray}H^{\prime\prime} (x)+\left[\displaystyle \frac{\gamma }{x}+\theta -2\,x\right]\,H^{\prime} (x)+\left[-\displaystyle \frac{\chi }{x}+{\rm{\Theta }}\right]\,H(x)=0,\end{eqnarray}$ where $ \begin{eqnarray}\begin{array}{rcl} & & \gamma =1+2\,A=1+2\,j,\\ & & {\rm{\Theta }}=\zeta +{B}^{2}-2(1+A)=\zeta +\displaystyle \frac{{\theta }^{2}}{4}-2(1+j),\\ & & \chi =\eta +(1+2\,A)B=\eta -\displaystyle \frac{\theta }{2}(1+2\,j).\end{array}\end{eqnarray}$ Equation (23 ) is the biconfluent Heun’s differential equation [17, 36, 45, 46], where H(x) is the Heun polynomial function.
The above equation (23 ) can be solved by the Frobenius method, writing the solution as a power series expansion around the origin: $ \begin{eqnarray}H(x)=\sum _{i=0}^{\infty }{c}_{i}\,{x}^{i}.\end{eqnarray}$ Substituting the above power series solution (25 ) into equation (23 ), we obtain the following recurrence relation for the coefficients: $ \begin{eqnarray}\begin{array}{rcl}{c}_{n+2} & = & \displaystyle \frac{1}{(n+2)(n+2+2\,j)}\\ & & \times \left[\{\chi -\theta (n+1)\}\,{c}_{n+1}-({\rm{\Theta }}-2\,n){c}_{n}\right],\end{array}\end{eqnarray}$ where the various coefficients are $ \begin{eqnarray}\begin{array}{rcl}{c}_{1} & = & \left(\displaystyle \frac{\eta }{(1+2\,j)}-\displaystyle \frac{\theta }{2}\right){c}_{0},\\ {c}_{2} & = & \displaystyle \frac{1}{4(1+j)}\,[(\chi -\theta ){c}_{1}-{\rm{\Theta }}\,{c}_{0}].\end{array}\end{eqnarray}$
As we have written the function H (x) as a power series expansion around the origin in equation (25 ), then the relativistic bound states solution can be achieved by imposing that the power series expansion (25 ) becomes a polynomial of degree n. With reference to the recurrence relation (26 ), we can see that the power series expansion H(x) becomes a polynomial of degree n by imposing the following two conditions [6, 17, 47, 48]: $ \begin{eqnarray}\begin{array}{c}\begin{array}{rcl} & & {\rm{\Theta }}=2\,n,\,(n=1,2,3,4,\ldots )\\ & & {c}_{n+1}=0.\end{array}\end{array}\end{eqnarray}$
By analyzing the condition Θ = 2 n, we obtain the following equation for eigenvalue En,l: $ \begin{eqnarray}\begin{array}{rcl} & & \zeta +\displaystyle \frac{{\theta }^{2}}{4}-2(1+j)=2\,n\\ \Rightarrow & & {E}_{n,l}^{2}-2\,{\rm{\Omega }}\,{E}_{n,l}(n+1+{l}_{0}+\sqrt{{l}_{0}^{2}-{e}^{2}\,{\xi }_{c}^{2}})-{k}^{2}-{M}^{2}\\ & & -2\,\xi \,{{\rm{\Omega }}}^{2}=0.\end{array}\end{eqnarray}$ The relativistic energy levels associated with nth radial modes is $ \begin{eqnarray}\begin{array}{rcl}{E}_{n,l} & = & {\rm{\Omega }}\left(n+1+\displaystyle \frac{(l-{\rm{\Phi }})}{\alpha }+\sqrt{\displaystyle \frac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}\,{\xi }_{c}^{2}}\right)\\ & & \pm \sqrt{{{\rm{\Omega }}}^{2}{\left(n+1+\displaystyle \frac{(l-{\rm{\Phi }})}{\alpha }+\sqrt{\displaystyle \frac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}{\xi }_{c}^{2}}\right)}^{2}+{k}^{2}+{M}^{2}+2\,\xi \,{{\rm{\Omega }}}^{2}}.\end{array}\end{eqnarray}$ We can see that the energy levels (30 ) get modifies and depend on the cosmic string α, the Aharonov–Bohm magnetic flux ΦB, and the Coulombic potential constant ξc.
The normalized eigenfunction is given by $ \begin{eqnarray}\displaystyle {\psi }_{n,l}(x)=| N{| }_{n,l}\,{x}^{\sqrt{\tfrac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}{\xi }_{c}^{2}}}\,{{\rm{e}}}^{\tfrac{e{\xi }_{c}{\rm{\Omega }}}{{E}_{n,l}}x}\,{{\rm{e}}}^{-\tfrac{{x}^{2}}{2}}\,H(x),\end{eqnarray}$ where ${E}_{n,l}$ is given in equation (30 ) and $| N{| }_{n,l}$ in the normalization constant, which cannot be determined directly since the function H(x) is a power series polynomial of degree n. One can determine the individual normalization constant by setting n = 1, 2, 3, … into the function and using the orthonormal condition over the range $[0,\infty )$ .
Now, we impose additional recurrence condition ${c}_{n+1}=0$ to find the individual energy levels and the corresponding wave-functions one by one, as described in [47, 48]. For example, for n = 1, we have c2 = 0, which is implied in (27 ), where $ \begin{eqnarray}{c}_{1}=\displaystyle \frac{2}{(\chi -\theta )}\,{c}_{0}\Rightarrow \displaystyle \frac{\eta }{1+2\,j}-\displaystyle \frac{\theta }{2}=\displaystyle \frac{2}{(\chi -\theta )}\end{eqnarray}$ is a constraint on physical parameters.
Therefore, the ground state energy level for n = 1 is given by $ \begin{eqnarray}\begin{array}{l}{E}_{1,l}\,=\,{\rm{\Omega }}\left(2+\displaystyle \frac{(l-{\rm{\Phi }})}{\alpha }+\sqrt{\displaystyle \frac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}\,{\xi }_{c}^{2}}\right)\,\pm \sqrt{{{\rm{\Omega }}}^{2}{\left(2+\displaystyle \frac{(l-{\rm{\Phi }})}{\alpha }+\sqrt{\displaystyle \frac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}{\xi }_{c}^{2}}\right)}^{2}+{k}^{2}+{M}^{2}+2\,\xi \,{{\rm{\Omega }}}^{2}}.\end{array}\end{eqnarray}$ The corresponding ground state radial wave-function is given as $ \begin{eqnarray}\displaystyle \psi {\left(x\right)}_{1,l}=| N{| }_{1,l}\,{x}^{\sqrt{\tfrac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}{\xi }_{c}^{2}}}\,{{\rm{e}}}^{\tfrac{e{\xi }_{c}{\rm{\Omega }}}{{E}_{1,l}}x}\,{{\rm{e}}}^{-\tfrac{{x}^{2}}{2}}({c}_{0}+{c}_{1}\,x),\end{eqnarray}$ where ${E}_{1,l}$ is given in (33 ) and $ \begin{eqnarray}\displaystyle {c}_{1}=-2\,e\,{\xi }_{c}\,\sqrt{\displaystyle \frac{{\rm{\Omega }}}{{E}_{1,l}}}\,\left[1+\displaystyle \frac{\left(\tfrac{(l-{\rm{\Phi }})}{\alpha }-\tfrac{{E}_{1,l}}{{\rm{\Omega }}}\right)}{\left(1+2\,\sqrt{\tfrac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}\,{\xi }_{c}^{2}}\right)}\right]\,{c}_{0}.\end{eqnarray}$ The normalization constant corresponding to n = 1 can be determined by $ \begin{eqnarray}\displaystyle \begin{array}{l}| N{| }_{1,l}\\ =\,{\left({\displaystyle \int }_{0}^{\infty }{x}^{2\sqrt{\tfrac{{\left(l-{\rm{\Phi }}\right)}^{2}}{{\alpha }^{2}}-{e}^{2}{\xi }_{c}^{2}}}{{\rm{e}}}^{-(x-\tfrac{2e{\xi }_{c}{\rm{\Omega }}}{{E}_{1,l}})x}{\left({c}_{0}+{c}_{1}x\right)}^{2}{\rm{d}}x\right)}^{-\tfrac{1}{2}},\end{array}\end{eqnarray}$ where c1 is given in equation (35 ).
With the general form of the relativistic energy eigenvalue equation (30 ), we see that the ground state of the Klein–Gordon particle subject to an electromagnetic potentials is defined by the quantum number n = 1 instead of the quantum number n = 0.
Observe that the relativistic energy eigenvalue equation (30 ) is dependent on the Aharonov–Bohm geometric quantum phase [37]. This dependence of relativistic energy eigenvalue on geometric quantum phase gives rise to an analogous effect on the Aharonov–Bohm effect for bound states [37, 42–44]. Thus, we find that ${E}_{n,\bar{l}}({{\rm{\Phi }}}_{B}+{{\rm{\Phi }}}_{0})={E}_{n,\bar{l}\mp \tau }({{\rm{\Phi }}}_{B})$ where ${{\rm{\Phi }}}_{0}=\pm \tfrac{2\,\pi \,\alpha }{e}\,\tau $ with $\tau =1,2,3,\ldots $ and $\bar{l}=\tfrac{l}{\alpha }$ .
We can see in both expressions of energy eigenvalues that the z-component of the angular momentum l is shifted: $ \begin{eqnarray}{l}_{0}=\displaystyle \frac{1}{\alpha }(l-{\rm{\Phi }})\end{eqnarray}$ giving an effective angular momentum due both to the boundary condition, which states that the total angle around the string is 2 π α, and to minimal coupling with electromagnetic interactions.
Formula equation (37 ) suggests that when the particle circles the string, the wave-function changes according to $ \begin{eqnarray}{\rm{\Psi }}\to {\rm{\Psi }}^{\prime} ={{\rm{e}}}^{2{\rm{i}}\pi {l}_{0}}\,{\rm{\Psi }}=\mathrm{Exp}\left\{\displaystyle \frac{2\,\pi \,{\rm{i}}}{\alpha }\left(l-\displaystyle \frac{e\,{{\rm{\Phi }}}_{B}}{2\,\pi }\right)\right\}\,{\rm{\Psi }}.\end{eqnarray}$ An immediate consequence of equation (37 ) is that the angular momentum operator may be redefined as $ \begin{eqnarray}{\hat{l}}_{0}=-\displaystyle \frac{{\rm{i}}}{\alpha }\left({\partial }_{\phi }-{\rm{i}}\,\displaystyle \frac{e\,{{\rm{\Phi }}}_{B}}{2\,\pi }\right),\end{eqnarray}$ where the additional term $-\tfrac{e\,{{\rm{\Phi }}}_{B}}{2\,\pi \,\alpha }$ takes into account the Aharonov–Bohm magnetic flux ΦB (internal magnetic field).
3. Conclusions
In this paper, we have presented a detailed study on the ground state of a bosonic massive charged particle in a Gödel-type space–time with interactions. By means of minimal substitution, we have introduced an electromagnetic four-vector potential into the Klein–Gordon equation. In case 1 of section 2 , we have considered zero potential with non-zero Aharonov–Bohm magnetic flux (ΦB) as the angular component of the electromagnetic four-vector potential, and derived the final form of the radial wave equation. We have then solved it using the Nikiforov–Uvarov method, and obtained a relativistic energy eigenvalue equation (13 ) and a corresponding normalized eigenfunction equation (14 ). We have seen that the relativistic energy eigenvalue equation (13 ) for ${{\rm{\Phi }}}_{B}\to 0$ and $\xi \to 0$ reduces to the result obtained in [9] and subsequently to the results in [3] provided $\alpha \to 1$ . In case 2 of section 2 , we have considered both Coulomb potential and the Aharonov–Bohm magnetic quantum flux. The radial equation of motion has derived by means of an appropriate ansatz and it has shown that such equation can be expressed in terms of the biconfluent Heun’s differential equation. We have solved this equation using a power series expansion around the origin before finally imposing a two conditions equation (28 ) to terminate the power series solution with a polynomial of degree n. Using the condition equation (28 ), we have obtained the energy eigenvalue equation (30 ), and have seen that the relativistic energy eigenvalue is modified, even though there is no direct interaction between the particles and the external magnetic fields. By imposing the additional recurrence condition ${c}_{n+1}=0$ , we have obtained the ground state energy level equation (33 ), and corresponding ground state radial wave-function equation (34 ) with equation (35 ) for the quantum number n = 1. The expression for the ground state energy level reveals the possibility of establishing a quantum condition between the energy of a bosonic massive charged particle and the parameter Ω that characterizes the vorticity parameter of the space–time study here. We have seen in both cases that the relativistic energy eigenvalue is a periodic function of the geometric quantum phase [37], that is, ${E}_{n,\bar{l}}({{\rm{\Phi }}}_{B}+{{\rm{\Phi }}}_{0})={E}_{n,\bar{l}\mp \tau }({{\rm{\Phi }}}_{B})$ where, ${{\rm{\Phi }}}_{0}=\pm \tfrac{2\,\pi \,\alpha }{e}\,\tau $ with τ = 1, 2, … and $\bar{l}=\tfrac{l}{\alpha }$ . This dependence of the relativistic energy eigenvalue on the geometric quantum phase gives rise to an analog effect on the Aharonov–Bohm effect for bound states [42–44]. In addition, in both the cases under discussion, we have seen that the angular momentum quantum number l is shifted, that is, $l\to {l}_{0}=\tfrac{1}{\alpha }(l-{\rm{\Phi }})$ , an effective angular quantum number. For $\alpha \to 1$ , the change in angular momentum quantum number, ${\rm{\Delta }}l=l-{l}_{0}=\tfrac{e\,{{\rm{\Phi }}}_{B}}{2\,\pi }$ is directly proportional to the magnetic quantum flux ΦB.
Therefore, in this paper, we have shown some results which, combined with results previously obtained in [3, 9], demonstrate many interesting possibilities. This is a fundamental subject in physics, and the connections between these theories (gravitation and quantum mechanics) are not well understood.