1. Introduction
The Schrödinger equation stands as one of the cornerstone equations in quantum physics [1, 2], serving as a fundamental tool for describing various quantum phenomena in both physical and chemical realms. Its solution is pivotal in non-relativistic quantum mechanics, as it reveals the wave function of a system, providing comprehensive insight into its physical properties. Solutions to the Schrödinger equation are sought for two primary types of potentials: central and non-central potentials. Central potentials hold significant utility in describing a wide array of quantum phenomena spanning molecular chemistry, quantum chemistry, atomic and molecular physics, astrophysics, particle physics, plasma physics and solid-state physics, among other areas of modern physics. Extensive research has been conducted in this domain, with numerous works dedicated to the study of central potentials. Notably, works such as [3–7] delve into the Coulomb potential, while [7, 8] tackle central potentials such as the Hulthen potential, Morse potential and harmonic oscillator. In addition [9–12], explore the pseudoharmonic potential, while [7, 11–15] investigate Mie-type or Kratzer potential. Furthermore, works such as [16, 17] focus on calculating Green's functions for specific systems, such as a quantum disk for the Helmholtz problem or a piecewise continuous symmetric spherical potential. Studies on delta potential [18] and Cornell potential [19, 20] also contribute to the understanding of central potentials. In recent years, non-central potentials have garnered significant attention in quantum mechanics due to their ability to provide a more comprehensive description of quantum systems. These potentials account for the dynamical properties of the system, encompassing both radial and angle-dependent components. This broader scope has led to increased interest and research activity in this area among scientists and researchers.
Various methods have been employed to solve the Schrödinger equation for both central and non-central potentials. Often, the Schrödinger equation yields differential equations of the hypergeometric type, with solutions expressed in terms of special functions. One commonly used method in such cases is the Nikiforov–Uvarov method [21]. For instance, in works such as [7, 22], the Schrödinger equation is tackled for a range of central and non-central potentials. In addition, alternative methods have been utilized to solve the Schrödinger equation, including the path integral approach [23]. Barut et al [5] explored the hydrogen atom problem in a spherical space, while Chetouani et al [24] computed the Green function for ring-shaped potentials. The algebraic method presents another avenue for addressing Schrödinger's problems. For example, Setare and Karimi [15] determined the energy eigenvalues and corresponding eigenfunctions of the Schrödinger equation for the Kratzer potential, whereas Chetouani et al [25] derived solutions for the Schrödinger equation considering potentials such as the Coulomb potential, the Hartmann ring-shaped potential and Smorodinsky–Winternitz potentials. The Laplace transform has been employed to determine exact solutions of the Schrödinger equation as demonstrated by Arda and Sever [26, 27], who derived bound state solutions for certain types of ring-shaped and Morse-like potentials. Asymptotic iteration has also been used to solve the Schrödinger equation for some central potentials [14], as well as the Klein–Gordon and Schrödinger equations for non-central potentials [28, 29]. These diverse methodologies contribute to the comprehensive understanding of quantum systems and their solutions.
The Schrödinger equation is exactly solvable for only a limited number of potentials, such as the Coulomb potential, harmonic oscillator, pseudoharmonic potential, among others. However, for many potentials, exact solutions remain elusive and approximations are necessary. In recent years, the discovery of a new class of quantum mechanical spectral problems, known as quasi-exactly solvable (QES) problems, has sparked significant interest among researchers [30–34]. These QES problems are characterized by the presence of a hidden algebraic structure in their Hamiltonians, albeit lacking hidden symmetry properties. They represent an intermediate scenario between exactly solvable and non-solvable problems. In QES problems, when one of the parameters assumes a specific value, a finite number of eigenstates can be determined, while the remaining states remain unknown. This indicates that the corresponding QES Schrödinger operator possesses a finite-dimensional invariant subspace. Exact solutions to QES problems offer valuable insight into the genuine and concealed properties of the physical systems under investigation. Frequently, these solutions reveal unexpected features and provide deeper understanding. Recent research on QES problems has utilized various methods, including the extended Nikiforov–Uvarov method, functional Bethe ansatz method and polynomial solutions of the biconfluent Heun equation [35–40]. These approaches contribute to the exploration and elucidation of QES problems, shedding light on their intriguing characteristics and implications in quantum mechanics.
Another type of problem more relevant consists of studying the Schrödinger equation in a curved space in the presence of topological defects [41]. Topological defects can appear in the contexts of gravitation and condensed matter physics. In gravitation, topological defects are associated with the evolutionary process of the early universe, in which symmetry-breaking phase transitions took place; and in the second, topological defects can appear during the synthesis of materials [42, 43]. In recent decades, the study of the physical effects of these topological defects on the physical properties of a system has been the subject of several very active research works [44–52].
This paper is a part of other publications focusing on QES problems [37, 39]. In [37], one of us together with a co-author present mathematical techniques and methods to solve quantum mechanical problems for certain non-central potentials. Specifically, we calculate the exact solutions of the Schrödinger equation using the QES method under a modified ring-shaped harmonic oscillator potential. By employing a change of variables in cylindrical coordinates and utilizing the method of separation of variables, we derive explicit expressions for the bound states and their associated eigenvalues of energy. These solutions are obtained using solutions of the biconfluent Heun equation. In [39], we extend our investigation to the bound states and corresponding energy eigenvalues of the Dirac equation involving non-central scalar and vector potentials. Specifically, we consider a modified double ring-shaped generalized Cornell potential within the framework of QES problems. Utilizing the method of separation of variables, we compute the angular parts of the solutions for the corresponding Schrödinger-like equation via the functional Bethe ansatz. Subsequently, the radial part is determined by solving the biconfluent Heun differential equation. These studies contribute to the advancement of understanding and solving quantum mechanical problems, particularly for systems characterized by non-central potentials. By employing rigorous mathematical techniques and leveraging the principles of QES problems, we elucidate the bound states and energy spectra associated with these complex quantum systems.
In our current study, we concentrate on examining the behavior of non-relativistic quantum particles interacting with an angular-dependent ring-shaped potential in the presence of topological defect space-times produced by a cosmic string and a point-like global monopole. We derive the radial and angular equations of the Schrödinger wave equation in both space-time geometries and determine the analytical eigenvalue solutions through special functions. Specifically, we demonstrate that the presence of a topological defect leads to modifications in the energy levels and the radial wave function compared to the results obtained in the flat space background. The presence of these defects disrupts the degeneracy of the energy spectra. Our study is structured as follows. In section 2 , we explicitly formulate the Schrödinger equation for an extended ring-shaped potential and decompose the wave equation into its angular and radial components. We obtain solutions for the angular parts employing techniques similar to those adopted in [37], and for the radial part using special functions to derive the eigenvalue solution. In section 3 , we provide numerical values for the eigenvalue solutions associated with the modes {n, ℓ, m} and analyze the outcomes. Finally, in section 4 , we present our conclusions. Throughout our analysis, we adopt a system of units where ℏ = c = G = 1 for consistency and convenience.
2. Cosmic string effect on the solution of the Schrödinger equation with extended ring-shaped potential
In this section, we investigate the behavior of non-relativistic quantum particles interacting with angular-dependent potential models in the presence of topological defects induced by a cosmic string. The geometry describing this topological defect in 4D is given by ${d{s}}_{4{\rm{D}}}^{2}=-{{\mathtt{d}}{t}}^{2}+{{\mathtt{d}}{r}}^{2}+{{r}}^{2}(d{\theta }^{2}\,+{\alpha }^{2}\,{\sin }^{2}\theta \,d{\varphi }^{2})=-{{\mathtt{d}}{t}}^{2}+{g}_{ij}\,{{\mathtt{d}}{x}}^{i}\,{{\mathtt{d}}{x}}^{j}$, where ‘α' represents the cosmic string parameter, and i, j = 1, 2, 3. This geometric setup has recently garnered significant attention in the investigation of quantum systems, both in the relativistic and non-relativistic limits, by numerous researchers (see, for example, [44–52]).
We initiate this study by formulating the time-independent Schrödinger wave equation as follows:
$\begin{eqnarray}\left[-\displaystyle \frac{1}{2\,M}\,\displaystyle \frac{1}{\sqrt{g}}\,{\partial }_{i}\left(\sqrt{g}\,{g}^{{ij}}\,{\partial }_{j}\right)+V({\boldsymbol{r}})\right]\,{\rm{\Psi }}({\boldsymbol{r}})=E\,{\rm{\Psi }}({\boldsymbol{r}}),\end{eqnarray}$
where $\Psi$(r) is the wave function, E is the particle's non-relativistic energy and V(r) is the potential given in the following form: $\begin{eqnarray}V({\boldsymbol{r}})=V(r,\theta )=\displaystyle \frac{{ \mathcal V }(\theta )}{{r}^{2}}.\end{eqnarray}$
Here, gij is the metric tensor of ${{\mathtt{d}}{s}}_{3{\rm{D}}}^{2}={g}_{ij}\,{{\mathtt{d}}{x}}^{{i}}\,{{\mathtt{d}}{x}}^{{j}}={r}^{2}(d{\theta }^{2}+{\alpha }^{2}\,{\sin }^{2}\theta \,d{\varphi }^{2})$ given by, $\begin{eqnarray}{g}_{{ij}}=\left(\begin{array}{rcl}1 & 0 & \,0\\ 0 & {r}^{2}\, & \,0\\ 0 & 0 & {\alpha }^{2}\,{r}^{2}\,{\sin }^{2}\theta \end{array}\right),\quad {g}^{{ij}}=\left(\begin{array}{rcl}1 & 0 & \,0\\ 0 & \displaystyle \frac{1}{{r}^{2}} & \,0\\ 0 & 0 & \displaystyle \frac{1}{{\alpha }^{2}\,{r}^{2}\,{\sin }^{2}\theta }\end{array}\right),\end{eqnarray}$
with the determinant $\begin{eqnarray}g={\mathsf{\det }}({g}_{{ij}})={\alpha }^{2}\,{r}^{4}\,{\sin }^{2}\theta .\end{eqnarray}$
In the spherical coordinates $\left(r,\theta ,\varphi \right)$, where r is the radius, θ is the polar angle and φ the azimuthal one, the Schrödinger equation takes the following form:
$\begin{eqnarray}\left[-\displaystyle \frac{1}{2\,M}\,{{\rm{\nabla }}}_{\mathrm{LB}-\mathrm{cosmic}-\mathrm{string}}^{2}+V(r,\theta ,\varphi )-E\right]\,{\rm{\Psi }}\left(r,\theta ,\varphi \right)=0,\end{eqnarray}$
where the Laplacian in topological defect is defined by, $\begin{eqnarray}\begin{array}{l}{{\rm{\nabla }}}_{\mathrm{LB}-\mathrm{cosmic}-\mathrm{string}}^{2}=\displaystyle \frac{1}{{r}^{2}}\,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}{r}}\left({r}^{2}\,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}{r}}\right)\\ \,+\displaystyle \frac{1}{{r}^{2}\,\sin \theta }\,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}\theta }\left(\sin \theta \,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}\theta }\right)+\displaystyle \frac{1}{{\alpha }^{2}\,{r}^{2}{\sin }^{2}\theta }\,\displaystyle \frac{{{\mathsf{d}}}^{2}}{{\mathsf{d}}{\phi }^{2}}.\end{array}\end{eqnarray}$
The potential V(r, θ) is given in equation (2 ) with,
$\begin{eqnarray}\begin{array}{l}{ \mathcal V }(\theta )=\displaystyle \frac{{b}_{0}+{b}_{1}\sin \theta +{b}_{2}{\sin }^{2}\theta +{b}_{3}{\sin }^{3}\theta +{b}_{4}{\sin }^{4}\theta }{{\cos }^{2}\theta {\sin }^{2}\theta }\\ \,+\displaystyle \frac{{b}_{5}+{b}_{6}{\cos }^{2}\theta +{b}_{7}{\cos }^{4}\theta }{{\cos }^{2}\theta {\sin }^{2}\theta }+\displaystyle \frac{{b}_{8}+{b}_{9}\sin \theta }{\sin \theta },\end{array}\end{eqnarray}$
where bi, i = 0, 1,…,9, are real constants.If ${ \mathcal V }(\theta )\ne 0$, many potentials are included in our potential: the ring-shaped potential [24, 53–57], the double ring-shaped potential, which is treated in many works [27–29, 54, 56, 58], the novel angle-dependent potential and the harmonic novel angle-dependent potential, which are studied using the NU method in [22, 59].
The total wave function is always expressible in terms of different variables as follows:
$\begin{eqnarray}{\rm{\Psi }}(r,\theta ,\varphi )={r}^{-1/2}{ \mathcal R }\left(r\right){\rm{\Theta }}(\theta ){\rm{\Phi }}(\varphi ).\end{eqnarray}$
Separating variables to equation (5 ) and using equations (6 )–(8 ), we obtain the following differential equations for angular coordinates and radial coordinate as follows:
$\begin{eqnarray}\left(\displaystyle \frac{{{\mathsf{d}}}^{2}}{{\mathsf{d}}{\varphi }^{2}}+{{\ell }}^{2}\right){\rm{\Phi }}(\varphi )=0,\end{eqnarray}$
$\begin{eqnarray}\left[\displaystyle \frac{{{\mathsf{d}}}^{2}}{{\mathsf{d}}{\theta }^{2}}+\cot \theta \,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}\theta }+\lambda -\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}\,{\sin }^{2}\theta }-2M{ \mathcal V }(\theta )\right]{\rm{\Theta }}(\theta )=0,\end{eqnarray}$
$\begin{eqnarray}\left[\displaystyle \frac{{{\mathsf{d}}}^{2}}{{{\mathsf{d}}{r}}^{2}}+\displaystyle \frac{1}{r}\,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}{r}}+2\,ME-\displaystyle \frac{(\lambda +1/4)}{{r}^{2}}\right]\,{ \mathcal R }(r)=0,\end{eqnarray}$
where ℓ2 and λ are called the separation constants.It is well known that the solution of the azimuthal part, i.e. solution of equation (9 ), with the boundary conditions ${\rm{\Phi }}\left(\varphi +2\pi \right)={\rm{\Phi }}\left(\varphi \right)$, is,
$\begin{eqnarray}{\rm{\Phi }}\left(\varphi \right)={{\mathsf{e}}}^{{\rm{i}}\,{\ell }\,\varphi },\end{eqnarray}$
where ℓ is an integer called the magnetic quantum number.Substituting potential ${ \mathcal V }(\theta )$ of expression (7 ) in equation (10 ), we obtain the following:
$\begin{eqnarray}\begin{array}{l}\left(\displaystyle \frac{{{\mathsf{d}}}^{2}}{{\mathsf{d}}{\theta }^{2}}+\cot \theta \,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}\theta }\right){\rm{\Theta }}(\theta )+\left[\lambda -\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}\,{\sin }^{2}\theta }-2\,M\left\{\displaystyle \frac{{b}_{0}+{b}_{1}\,\sin \theta +{b}_{2}\,{\sin }^{2}\theta +{b}_{3}\,{\sin }^{3}\theta +{b}_{4}\,{\sin }^{4}\theta }{{\cos }^{2}\theta \,{\sin }^{2}\theta }\right.\right.\\ \left.\left.+\displaystyle \frac{{b}_{5}+{b}_{6}\,{\cos }^{2}\theta +{b}_{7}\,{\cos }^{4}\theta }{{\cos }^{2}\theta \,{\sin }^{2}\theta }+\displaystyle \frac{{b}_{8}+{b}_{9}\,\sin \theta }{\sin \theta }\right\}\right]\,{\rm{\Theta }}(\theta )=0.\end{array}\end{eqnarray}$
We change the variable $x=\sin \theta $, and equation (13 ) becomes:
$\begin{eqnarray}\begin{array}{l}\left[(1-{x}^{2})\displaystyle \frac{{{\mathsf{d}}}^{2}}{{{\mathsf{dx}}}^{2}}+\displaystyle \frac{1-2\,{x}^{2}}{x}\displaystyle \frac{{\mathsf{d}}}{{\mathsf{dx}}}\right]\\ \times \ {\rm{\Theta }}(x)+\left[\lambda -\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}\,{x}^{2}}-2\,M\left(\displaystyle \frac{{b}_{0}+{b}_{5}+{b}_{1}x+{b}_{2}{x}^{2}+{b}_{3}{x}^{3}+{b}_{4}{x}^{4}}{{x}^{2}\left(1-{x}^{2}\right)}\right.\right.\\ \left.\left.+\displaystyle \frac{\left({b}_{6}+{b}_{7}\right)+{b}_{8}x+\left({b}_{9}-{b}_{7}\right){x}^{2}}{{x}^{2}}\right)\right]{\rm{\Theta }}(x)=0.\end{array}\end{eqnarray}$
If we impose the following constraints on parameters,14 ) takes the following form form:
$\begin{eqnarray}\left\{\begin{array}{l}{b}_{1}+{b}_{3}=0\ ;\\ {b}_{0}+{b}_{2}+{b}_{4}+{b}_{5}=0,\end{array}\right.\end{eqnarray}$
the differential equation ( $\begin{eqnarray}\begin{array}{l}\left[{x}^{2}(1-{x}^{2})\frac{{{\mathsf{d}}}^{2}}{{{\mathsf{dx}}}^{2}}+x(1-2\,{x}^{2})\frac{{\mathsf{d}}}{{\mathsf{dx}}}\right]\,{\rm{\Theta }}(x)\\ \,+\left[{c}_{0}-\frac{{\ell }^{2}}{{\alpha }^{2}}+{c}_{1}x+\left({c}_{2}+\lambda \right){x}^{2}\right]{\rm{\Theta }}(x)=0,\end{array}\end{eqnarray}$
where $\begin{eqnarray}\left\{\begin{array}{l}{c}_{0}=2\,M\left({b}_{2}+{b}_{4}-{b}_{6}-{b}_{7}\right);\\ {c}_{1}=-2\,M\left({b}_{1}+{b}_{8}\right);\\ {c}_{2}=-2\,M\left({b}_{9}-{b}_{7}-{b}_{4}\right).\end{array}\right.\end{eqnarray}$
Following the Bethe ansatz method [35, 60], one of us obtained the angular solution of the above angular differential equation in [61]. Equation (1 ) admits polynomial solutions of degree m and the values of separation constant λ with a few constraints on various parameters satisfy the following constraints:
$\begin{eqnarray}\lambda ={\lambda }_{{\ell },m}=m(m+1)+2\,M({b}_{9}-{b}_{7}-{b}_{4}),\end{eqnarray}$
$\begin{eqnarray}M({b}_{1}+{b}_{8})=-m\,\displaystyle \sum _{i=1}^{m}{x}_{i},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}M({b}_{2}+{b}_{4}-{b}_{6}-{b}_{7})\\ \,=m\,\displaystyle \sum _{i=1}^{m}\,{x}_{i}^{2}+\displaystyle \sum _{i\lt j}^{m}\,{x}_{i}\,{x}_{j}+\displaystyle \frac{{{\ell }}^{2}-{\alpha }^{2}\,{m}^{2}}{2\,{\alpha }^{2}},\end{array}\end{eqnarray}$
where x1, x2,…,xm are distinct roots of the polynomial solutions that satisfy, in this case, the following Bethe ansatz equations: $\begin{eqnarray}\displaystyle \sum _{j\ne i}^{m}\displaystyle \frac{2}{{x}_{i}-{x}_{j}}+\displaystyle \frac{2{x}_{i}^{2}-1}{{x}_{i}\left({x}_{i}^{2}-1\right)}=0.\qquad (i=1,2,\ldots ,m)\end{eqnarray}$
Now, we solve the radial equation (11 ). We can write the radial equation (11 ) in the following form:18 ).
$\begin{eqnarray}\left[\displaystyle \frac{{{\mathsf{d}}}^{2}}{{{\mathsf{d}}{r}}^{2}}+\displaystyle \frac{1}{r}\,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}{r}}+2\,ME-\displaystyle \frac{{\iota }^{2}}{{r}^{2}}\right]\,{ \mathcal R }(r)=0,\end{eqnarray}$
where $\iota =\sqrt{\lambda +1/4}$ and λ is given in (Equation (19 ) is the second-order Bessel equation whose exact solutions are well known [62, 63]. We are mainly interested in the solution that is regular at the origin, r = 0. Now, the solution of equation (19 ) is $R(r)={f}_{1}\,{J}_{\iota }(\sqrt{2\,ME}\,r)+{f}_{2}\,{Y}_{\iota }(\sqrt{2\,ME}\,r)$, where Jι and Yι are the Bessel functions of the first and second kind, respectively. It is also well known that the Bessel function of the second kind, Yι, diverges at the origin r = 0 whereas the first kind is finite. Employing the wave function requirement R(r → 0) → 0 results in f2 = 0. Therefore, the regular solution is given by,
$\begin{eqnarray}{ \mathcal R }(r)={f}_{1}\,{J}_{\iota }\left(\sqrt{2\,ME}\,r\right),\end{eqnarray}$
where f1 is a constant.To obtain solution of the quantum system under investigation, we introduce a hard-wall confining potential condition, which states that at some finite radial distance r = r0, the radial wave function R vanishes, that is,
$\begin{eqnarray}{ \mathcal R }(r={r}_{0})=0.\end{eqnarray}$
This confinement is particularly significant since it offers a robust approximation for investigating the quantum properties of systems, such as gas molecules and other particles inherently confined within a defined spatial domain. The study of the hard-wall confinement potential has proved to be valuable in various contexts. For instance, it has been examined in the context of scalar fields under non-inertial effects [64], studies of the Klein–Gordon oscillator field under cosmic string effects [65], investigations of the Dirac neutral particle [66–68], analyses of the harmonic oscillator in an elastic medium [69], analogous to the Landau–Aharonov–Casher system in cosmic string space-time [70], and exploration into the behavior of the Dirac oscillator [71] and Klein–Gordon oscillator fields [71, 72]. This exploration of the hard-wall potential across diverse scenarios enriches our understanding of its impact on quantum systems, offering insight into the behavior of quantum particles subject to this form of confinement. To extract energy levels from the boundary conditions, we fix r = r0 to be sufficiently large, ensuring that $\sqrt{2\,ME}\,{r}_{0}\gg 1$.The asymptotic form of the Bessel function of the first kind is given by,
$\begin{eqnarray}{J}_{\iota }(\sqrt{2\,ME}\,r)\propto \cos \left(\sqrt{2\,ME}\,r-\displaystyle \frac{\pi }{2}\,\iota -\displaystyle \frac{\pi }{4}\right).\end{eqnarray}$
Substituting equation (23 ) into equation (20 ) and using condition (22 ), we find,
$\begin{eqnarray}\cos \left(\sqrt{2\,ME}\,{r}_{0}-\displaystyle \frac{\pi }{2}\,\iota -\displaystyle \frac{\pi }{4}\right)=0.\end{eqnarray}$
Simplification of the above relation results in energy eigenvalues of quantum particles associated with the mode {ℓ, m} given by, $\begin{eqnarray}\begin{array}{rcl}{E}_{{\ell },m,n} & = & \displaystyle \frac{1}{2\,M}\left(n+\displaystyle \frac{1}{2}\sqrt{{\left(m+\displaystyle \frac{1}{2}\right)}^{2}+2\,M({b}_{9}-{b}_{7}-{b}_{4})}\right.\\ & & +\ {\left.\displaystyle \frac{3}{4}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{array}\end{eqnarray}$
where n = 0, 1, 2, ….Then, the wave function is as follows:16 ) and Cℓ,m,n are normalization constants, for ∣ℓ∣, m, n = 0, 1, 2,….
$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },m,n}\left(r,\theta ,\varphi \right)={C}_{{\ell },m,n}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },m}}\\ \,\times \,(\sqrt{2\,M{E}_{{\ell },m,n}}\,r){{\rm{\Theta }}}_{m,{\ell }}(\theta )\exp \left({\rm{i}}\,{\ell }\,\varphi \right),\end{array}\end{eqnarray}$
where ${\iota }_{{\ell },m}=\sqrt{{\lambda }_{{\ell },m}+1/4}$ , Θm,ℓ(θ) is the angular solution of the differential equation (2.1. Numerical examples of the eigenvalue solution
In this section, we give a particular example of potential (7 ) for which we apply all results, found in the previous sections, to calculate the bound states as well as the corresponding energy levels for some values of ℓ, m, n.
In this particular case and for more simplicity in the calculation we take M = 1, , b1 = − 1, b9 = b4 = b6 − b2, and given the conditions (15 ) we have b3 = 1, b0 + b2 + b4 + b5 = 0.
Then, the potential (2 ) reduces to:18 )–(20 ) will reduce to the following constraints:21 ).
$\begin{eqnarray}V(r,\theta )=\displaystyle \frac{{ \mathcal V }(\theta )}{{r}^{2}},\end{eqnarray}$
and ${ \mathcal V }(\theta )$ is given as, $\begin{eqnarray}{ \mathcal V }(\theta )={b}_{7}{\cot }^{2}\theta +\displaystyle \frac{{b}_{8}-1}{\sin \theta }.\end{eqnarray}$
Then, the constraints ( $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{\ell ,m}=m\left[1+2\displaystyle \sum _{i=1}^{m}{x}_{i}^{2}\right]+2\displaystyle \sum _{i\lt j}^{m}{x}_{i}{x}_{j}+\displaystyle \frac{{\ell }^{2}}{{\alpha }^{2}};\\ {b}_{8}=1-m\displaystyle \sum _{i=1}^{m}{x}_{i};\\ {b}_{7}=\displaystyle \frac{{\alpha }^{2}{m}^{2}-{\ell }^{2}}{2{\alpha }^{2}}-m\displaystyle \sum _{i=1}^{m}{x}_{i}^{2}-\displaystyle \sum _{i\lt j}^{m}{x}_{i}{x}_{j},\end{array}\right.\end{eqnarray}$
so that the roots x1, x2,…,xm satisfy the Bethe ansatz equations (From the relation (23 ), we know that the radial solution is given as,
$\begin{eqnarray}{{ \mathcal R }}_{{\ell },m,n}(r)={f}_{1}\,{J}_{{\iota }_{{\ell },m}}(\sqrt{2\,{E}_{{\ell },m,n}}\,r),\end{eqnarray}$
where f1 is a constant (chosen here to be unity for simplicity), and the energy eigenvalues Eℓ,m,n are now given by the following expression: $\begin{eqnarray}{E}_{{\ell },m,n}=\displaystyle \frac{1}{2}{\left(n+\displaystyle \frac{1}{2}\sqrt{{\left(m+\displaystyle \frac{1}{2}\right)}^{2}-2\,{b}_{7}}+\displaystyle \frac{3}{4}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$
for ∣ℓ∣, m,n = 0,1,2,…The angular solution of the above angular differential equation (10 ) admits polynomial solutions of degree m and the values of separation constant λℓ,m with a few constraints on various parameters satisfy constraints (31 ). Then, in this case and for the constraints, we can express some polynomial solutions, in terms of $\sin \theta $, of equation (10 ) as,
| i | (i) For m = 0 is Θ0(θ) ≡ 1 with the constraints: $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{{\ell },0}=\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}};\\ {b}_{8}=1;\\ {b}_{7}=-\displaystyle \frac{{{\ell }}^{2}}{2{\alpha }^{2}}.\end{array}\right.\end{eqnarray}$ |
| ii | (ii) For m = 1, we have ${x}_{1}=\pm \displaystyle \frac{\sqrt{2}}{2}$ then we get: |
| • | ${{\rm{\Theta }}}_{1}(\theta )=\sin \theta +\displaystyle \frac{\sqrt{2}}{2},$ for ${x}_{1}=-\displaystyle \frac{\sqrt{2}}{2}$, the constraints are, $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{{\ell },1}=2+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}};\\ {b}_{8}=1+\displaystyle \frac{1}{\sqrt{2}};\\ {b}_{7}=-\displaystyle \frac{{{\ell }}^{2}}{2{\alpha }^{2}}.\end{array}\right.\end{eqnarray}$ |
| • | ${{\rm{\Theta }}}_{1}(\theta )=\sin \theta -\displaystyle \frac{\sqrt{2}}{2},$ for ${x}_{1}=\displaystyle \frac{\sqrt{2}}{2}$ and the conditions are, $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{{\ell },1}=2+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}};\\ {b}_{8}=1-\displaystyle \frac{1}{\sqrt{2}};\\ {b}_{7}=-\displaystyle \frac{{{\ell }}^{2}}{2{\alpha }^{2}}.\end{array}\right.\end{eqnarray}$ |
| i | (iii) For m = 2 $\begin{eqnarray}{{\rm{\Theta }}}_{2}(x)=(x-{x}_{1})(x-{x}_{2}),\end{eqnarray}$ where the roots x1 , x2 satisfy the following Bethe ansatz equations: $\begin{eqnarray}\displaystyle \frac{2}{{x}_{1}-{x}_{2}}+\displaystyle \frac{2{x}_{1}^{2}-1}{{x}_{1}\left({x}_{1}^{2}-1\right)}=0,\end{eqnarray}$ $\begin{eqnarray}\displaystyle \frac{2}{{x}_{2}-{x}_{1}}+\displaystyle \frac{2{x}_{2}^{2}-1}{{x}_{2}\left({x}_{2}^{2}-1\right)}=0.\end{eqnarray}$ After solving the previous nonlinear algebraic system, we obtain the following values of roots x1 and x2: $\begin{eqnarray}\left\{\begin{array}{l}{x}_{1}=-\sqrt{\displaystyle \frac{2}{3}}\ ,\ {x}_{2}=\sqrt{\displaystyle \frac{2}{3}};\\ {x}_{1}=\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}+\sqrt{11}\right),{x}_{2}=-\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}-\sqrt{11}\right);\\ {x}_{1}=-\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}+\sqrt{11}\right),{x}_{2}=\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}-\sqrt{11}\right).\end{array}\right.\end{eqnarray}$ Then the angular solution has one of the following polynomial solutions of degree 2, in $\sin \theta $, with the constraints ( |
| • | For ${x}_{1}=-\sqrt{\displaystyle \frac{2}{3}}$ , ${x}_{2}=\sqrt{\displaystyle \frac{2}{3}}$: ${{\rm{\Theta }}}_{2}(\theta )={\sin }^{2}\theta -\displaystyle \frac{2}{3},$ under the constraints: $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{{\ell },2}=6+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}};\\ {b}_{8}=1;\\ {b}_{7}=-\displaystyle \frac{{{\ell }}^{2}}{2{\alpha }^{2}}.\end{array}\right.\end{eqnarray}$ |
| • | For${x}_{1}=\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}+\sqrt{11}\right),{x}_{2}=-\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}-\sqrt{11}\right)$, we have ${{\rm{\Theta }}}_{2}(\theta )={\sin }^{2}\theta -\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4},$ with the constraints: $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{{\ell },2}=6+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}};\\ {b}_{8}=1-\displaystyle \frac{\sqrt{22}}{2};\\ {b}_{7}=-\displaystyle \frac{{{\ell }}^{2}}{2{\alpha }^{2}}.\end{array}\right.\end{eqnarray}$ |
| • | For ${x}_{1}=-\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}+\sqrt{11}\right),\ {x}_{2}=\displaystyle \frac{1}{4\sqrt{2}}\left(\sqrt{3}-\sqrt{11}\right)$, we obtain ${{\rm{\Theta }}}_{2}(\theta )={\sin }^{2}\theta +\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4}$ and the constraints are: $\begin{eqnarray}\left\{\begin{array}{l}{\lambda }_{{\ell },2}=6+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}};\\ {b}_{8}=1+\displaystyle \frac{\sqrt{22}}{2};\\ {b}_{7}=-\displaystyle \frac{{{\ell }}^{2}}{2{\alpha }^{2}}.\end{array}\right.\end{eqnarray}$ |
Using relations (28 ) , (33 ) and the above calculations we can give some explicit bound states ${{\rm{\Psi }}}_{{\ell },m,n}\left(r,\theta ,\varphi \right)$ and their energy levels Eℓ,m,n for m = 0, 1, 2, n = 0, 1, 2, 3.
| 1. For m = 0 |
| • | If n = 0, we get: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{0,0}}\left(r,\theta ,\varphi \right)\\ \,={C}_{{\ell },\mathrm{0,0}}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },0}}\left(\sqrt{2\,{E}_{{\ell },\mathrm{0,0}}}\,r\right)\,\exp \,\left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ and the associated energy level is $\begin{eqnarray}{E}_{{\ell },\mathrm{0,0}}=\displaystyle \frac{1}{8}{\left(\sqrt{\displaystyle \frac{1}{4}+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}}}+\displaystyle \frac{3}{2}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ where ${\iota }_{{\ell },0}=\sqrt{\tfrac{{{\ell }}^{2}}{{\alpha }^{2}}+\tfrac{1}{4}}$ and ∣ℓ∣ = 0, 1, 2, … with constraints ( |
| • | If n = 3, |
$\begin{eqnarray}{{\rm{\Psi }}}_{{\ell },\mathrm{0,3}}\left(r,\theta ,\varphi \right)={C}_{{\ell },\mathrm{0,3}}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },0}}(\sqrt{2\,{E}_{{\ell },\mathrm{0,3}}}\,r)\exp \left({\rm{i}}{\ell }\varphi \right),\end{eqnarray}$
and the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{0,3}}=\displaystyle \frac{1}{8}{\left(\sqrt{\displaystyle \frac{1}{4}+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}}}+\displaystyle \frac{15}{2}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$
for all ℓ, where ∣ℓ∣ = 0, 1, 2, … with the constraint (| 2. For m = 1 |
| • | If n = 1 and ${{\rm{\Theta }}}_{1}(\theta )=\sin \theta +\displaystyle \frac{\sqrt{2}}{2},$ we have: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{1,1}}\left(r,\theta ,\varphi \right)\\ \,={C}_{{\ell },\mathrm{1,1}}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },1}}(\sqrt{2\,M\,{E}_{{\ell },\mathrm{1,1}}}\,r)\left(\sin \theta +\displaystyle \frac{\sqrt{2}}{2}\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ and the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{1,1}}=\displaystyle \frac{1}{8}{\left(\sqrt{\displaystyle \frac{9}{4}+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}}}+\displaystyle \frac{7}{2}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ where ${\iota }_{{\ell },1}=\sqrt{\tfrac{{{\ell }}^{2}}{{\alpha }^{2}}+\tfrac{9}{4}}$ and ∣ℓ∣ = 0, 1, 2, … with constraints ( |
| • | If n = 2 and ${{\rm{\Theta }}}_{1}(\theta )=\sin \theta -\displaystyle \frac{\sqrt{2}}{2},$ we get: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{1,2}}\left(r,\theta ,\varphi \right)\\ \,={C}_{{\ell },\mathrm{1,2}}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },1}}\left(\sqrt{2\,M\,{E}_{{\ell },\mathrm{1,2}}}\,r\right)\left(\sin \theta -\displaystyle \frac{\sqrt{2}}{2}\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ and the associated energy level is $\begin{eqnarray}{E}_{{\ell },\mathrm{1,2}}=\displaystyle \frac{1}{8}{\left(\sqrt{\displaystyle \frac{9}{4}+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}}}+\displaystyle \frac{11}{2}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ where ${\iota }_{{\ell },1}=\sqrt{\tfrac{{{\ell }}^{2}}{{\alpha }^{2}}+\tfrac{9}{4}}$ and ∣ℓ∣ = 0, 1, 2, … with constraints ( |
| 3. For m = 2 |
| • | If n = 0 and ${{\rm{\Theta }}}_{2}(\theta )={\sin }^{2}\theta -\displaystyle \frac{2}{3},$ we get: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{2,0}}\left(r,\theta ,\varphi \right)\\ \,={C}_{{\ell },\mathrm{2,0}}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },2}}(\sqrt{2\,{E}_{{\ell },\mathrm{2,0}}}\,r)\left({\sin }^{2}\theta -\displaystyle \frac{2}{3}\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ and the associated energy level is $\begin{eqnarray}{E}_{{\ell },\mathrm{2,0}}=\displaystyle \frac{1}{8}{\left(\sqrt{\displaystyle \frac{25}{4}+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}}}+\displaystyle \frac{3}{2}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ where ${\iota }_{{\ell },2}=\sqrt{\tfrac{{{\ell }}^{2}}{{\alpha }^{2}}+\tfrac{25}{4}}$ and ∣ℓ∣ = 0, 1, 2, … with constraints ( |
| • | If n = 2 and ${{\rm{\Theta }}}_{2}(\theta )={\sin }^{2}\theta -\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4},$ we have: $\begin{eqnarray}{{\rm{\Psi }}}_{{\ell },\mathrm{2,2}}\left(r,\theta ,\varphi \right)={C}_{{\ell },\mathrm{2,2}}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },2}}(\sqrt{2\,{E}_{{\ell },\mathrm{2,2}}}\,r)\left({\sin }^{2}\theta -\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4}\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{eqnarray}$ and the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{2,2}}=\displaystyle \frac{1}{8}{\left(\sqrt{\displaystyle \frac{25}{4}+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}}}+\displaystyle \frac{11}{2}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ where ${\iota }_{{\ell },2}=\sqrt{\tfrac{{{\ell }}^{2}}{{\alpha }^{2}}+\tfrac{25}{4}}$ and ∣ℓ∣ = 0, 1, 2, … with constraints ( |
| • | If n = 3 and ${{\rm{\Theta }}}_{2}(\theta )={\sin }^{2}\theta +\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4},$ we obtain: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{2,3}}\left(r,\theta ,\varphi \right)={C}_{{\ell },\mathrm{2,3}}\ {r}^{-1/2}{J}_{{\iota }_{{\ell },2}}\\ \,\times \,(\sqrt{2\,{E}_{{\ell },\mathrm{2,3}}}\,r)\left({\sin }^{2}\theta +\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4}\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ and the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{2,3}}=\displaystyle \frac{1}{8}{\left(\sqrt{\displaystyle \frac{25}{4}+\displaystyle \frac{{{\ell }}^{2}}{{\alpha }^{2}}}+\displaystyle \frac{15}{2}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ where ${\iota }_{{\ell },2}=\sqrt{\tfrac{{{\ell }}^{2}}{{\alpha }^{2}}+\tfrac{25}{4}}$ and ∣ℓ∣ = 0, 1, 2, … with constraints ( |
Throughout this section of numerical computations, we have seen that the inclusion of a topological defect in the form of a cosmic string, characterized by the parameter α, exerts a significant influence on the energy levels and the corresponding wave functions of quantum particles. This parameter distinctly alters the eigenvalue solutions of the quantum systems, contrasting with outcomes derived within the backdrop of Minkowski flat space. Consequently, it disrupts the degeneracy observed in the energy spectra.
We have illustrated graphs depicting the energy levels Eℓ,m,n of equation (33 ) for the scenarios where m = 0 in figure 1 and m = 1 in figure 2, across various values of the topological parameter α and the quantum numbers {n, ℓ}. Similarly, we have plotted graphs representing the radial wave function ${{ \mathcal R }}_{{\ell },m,n}$ of equation (32 ) for m = 0 in figure 3 and m = 1 in figure 4, considering different values of the topological parameter α and the quantum numbers {n, ℓ}.
Figure 1. Energy levels Eℓ,m,n of equation ( |
Figure 2. Energy levels Eℓ,m,n of equation ( |
Figure 3. Radial wave function ${{ \mathcal R }}_{{\ell },m,n}$ of equation ( |
Figure 4. Radial wave function ${{ \mathcal R }}_{{\ell },m,n}$ of equation ( |
3. Global monopole effect on the solution of the Schrödinger equation with extended ring-shaped potential
In this section, we investigate the behavior of non-relativistic quantum particles interacting with angular-dependent potential models in the presence of topological defects induced by a point-like global monopole. The geometry describing this topological defect in 4D is given by ${{\mathsf{d}}{s}}_{{\mathsf{4}}{\mathsf{D}}}^{2}=-{d{t}}^{2}\,+\tfrac{{d{r}}^{2}}{{\tilde{\alpha }}^{2}}$ $+{r}^{2}(d{\theta }^{2}$ $+{\sin }^{2}\theta \,d{\varphi }^{2})$ $=-{{\mathsf{d}}{t}}^{2}+{{g}}_{{ij}}\,{{\mathsf{d}}{x}}^{{i}}\,{{\mathsf{d}}{x}}^{{j}}$, where $\tilde{\alpha }$ represents the global monopole parameter, and i, j = 1, 2, 3. This geometric setup has recently garnered significant attention in the investigation of quantum systems, both in the relativistic and non-relativistic limits, by numerous researchers (see, for example, [47, 71–87]).
The spatial metric tensor gij and its contravariant form for point-like global monopole geometry is given by,
$\begin{eqnarray}{g}_{{ij}}=\left(\begin{array}{lcc}\frac{1}{{\tilde{\alpha }}^{2}} & 0 & 0\\ 0 & {r}^{2}\, & \,0\\ 0 & 0 & {r}^{2}\,{\sin }^{2}\theta \end{array}\right),\quad {g}^{{ij}}=\left(\begin{array}{lll}{\tilde{\alpha }}^{2} & 0 & 0\\ 0 & \frac{1}{{r}^{2}} & 0\\ 0 & 0 & \frac{1}{{r}^{2}\,{\sin }^{2}\theta }\end{array}\right),\end{eqnarray}$
with the determinant, $\begin{eqnarray}g={\mathsf{\det }}({g}_{{ij}})=\displaystyle \frac{{r}^{4}\,{\sin }^{2}\theta }{{\tilde{\alpha }}^{2}}.\end{eqnarray}$
Therefore, the Schrödinger equation (1 ) using equations (58 ), (59 ) and (8 ) and separating the variables, we obtain the following differential equations:
$\begin{eqnarray}\left(\displaystyle \frac{{{\mathsf{d}}}^{2}}{{\mathsf{d}}{\varphi }^{2}}+{{\ell }}^{2}\right){\rm{\Phi }}(\varphi )=0,\end{eqnarray}$
$\begin{eqnarray}\left[\displaystyle \frac{{{\mathsf{d}}}^{2}}{{\mathsf{d}}{\theta }^{2}}+\cot \theta \,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}\theta }+\lambda -\displaystyle \frac{{{\ell }}^{2}}{{\sin }^{2}\theta }-2\,M\,{ \mathcal V }(\theta )\right]\,{\rm{\Theta }}(\theta )=0,\end{eqnarray}$
$\begin{eqnarray}\left[\displaystyle \frac{{{\mathsf{d}}}^{2}}{{{\mathsf{d}}{r}}^{2}}+\displaystyle \frac{1}{r}\,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{d}}{r}}+\displaystyle \frac{1}{{\tilde{\alpha }}^{2}}\left(2\,ME-\displaystyle \frac{(\lambda +1/4)}{{r}^{2}}\right)\right]\,{ \mathcal R }(r)=0,\end{eqnarray}$
where ℓ2 and λ are called the separation constants.The solutions of angular parts equations (60 )–(61 ) for the chosen ring-shaped potential (7 ) are discussed in [61]. However, the radial solution of the present study is different from the previous paper. Equation (62 ) can be expressed in the following form:61 ).
$\begin{eqnarray}\left[\displaystyle \frac{{{\mathsf{d}}}^{2}}{{{\mathsf{dr}}}^{2}}+\displaystyle \frac{1}{r}\,\displaystyle \frac{{\mathsf{d}}}{{\mathsf{dr}}}+{\bar{E}}^{2}-\displaystyle \frac{{\tau }^{2}}{{r}^{2}}\right]\,{ \mathcal R }(r)=0,\end{eqnarray}$
where $\bar{E}=\tfrac{\sqrt{2\,ME}}{\tilde{\alpha }}$, and $\tau =\tfrac{1}{\tilde{\alpha }}\sqrt{\lambda +1/4}$ with λ is the separation constant that can be obtained by solving the angular equation (Equation (63 ) is the Bessel differential equation whose solutions are well known. Following the similar procedure carried out in section 2 , one can find the energy eigenvalue of non-relativistic quantum particles as follows:
$\begin{eqnarray}\begin{array}{l}{E}_{{\ell },m,n}=\\ \,\displaystyle \frac{{\tilde{\alpha }}^{2}}{2\,M}{\left(n+\displaystyle \frac{1}{2\,\tilde{\alpha }}\,\sqrt{{\left(m+\displaystyle \frac{1}{2}\right)}^{2}+2\,M({b}_{9}-{b}_{7}-{b}_{4})}+\displaystyle \frac{3}{4}\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{array}\end{eqnarray}$
where n = 0, 1, 2, … and r0 is an arbitrary constant.The corresponding wave function is given by,61 ) and ${\tilde{C}}_{{\ell },m,n}$ is normalization constants for ∣ℓ∣, m, n = 0, 1, 2,….
$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },m,n}\left(r,\theta ,\varphi \right)\\ \,={\tilde{C}}_{{\ell },m,n}\ {r}^{-1/2}{J}_{{\tau }_{{\ell },m}}\left(\displaystyle \frac{\sqrt{2\,M{E}_{{\ell },m,n}}}{\tilde{\alpha }}\,r\right){\tilde{{\rm{\Theta }}}}_{m,{\ell }}(\theta )\exp \left({\rm{i}}\,{\ell }\varphi \right),\end{array}\end{eqnarray}$
where ${\tau }_{{\ell },m}=\tfrac{1}{\tilde{\alpha }}\,\sqrt{{\lambda }_{{\ell },m}+1/4}$, ${\tilde{{\rm{\Theta }}}}_{m,{\ell }}(\theta )$ is the angular solution of the differential equation (3.1. Numerical examples of the eigenvalue solution
As in the previous section, section 3 , and following a similar procedure, we give a particular example of potential (7 ) for which we apply all results, found above, to calculate the bound states as well as the corresponding energy levels for some values of ℓ, m, n.
For simplicity we take, in this particular case:
M = 1, , b0 = b1 = b3 = b5 = 0, b2 = − b4, b6 = b7 and b2 + b9 − b6 = 0.
Then, the potential (2 ) reduces to,21 ).
$\begin{eqnarray}V(r,\theta )=\displaystyle \frac{{ \mathcal V }(\theta )}{{r}^{2}},\end{eqnarray}$
and ${ \mathcal V }(\theta )$ is given as, $\begin{eqnarray}{ \mathcal V }(\theta )=\displaystyle \frac{2\ {b}_{6}}{{\sin }^{2}\theta }+\displaystyle \frac{{b}_{8}}{\sin \theta },\end{eqnarray}$
and the constraints will be as follows: $\begin{eqnarray}\lambda ={\lambda }_{{\ell },m}=m(m+1),\end{eqnarray}$
$\begin{eqnarray}{b}_{8}=-m\,\displaystyle \sum _{i=1}^{m}{x}_{i},\end{eqnarray}$
$\begin{eqnarray}{b}_{6}=-\displaystyle \frac{1}{2}\left(m\,\displaystyle \sum _{i=1}^{m}\,{x}_{i}^{2}+\displaystyle \sum _{i\lt j}^{m}\,{x}_{i}\,{x}_{j}+\displaystyle \frac{{{\ell }}^{2}-\,{m}^{2}}{2\,}\right),\end{eqnarray}$
where ∣ℓ∣, m = 0, 1, 2,…, and x1, x2,…,xm are distinct roots of the polynomial solutions satisfying the Bethe ansatz equations (It is clear that, in this case and from the previous relations, one can deduce that ιℓ,m = m + 1/2 and the energy eigenvalue is as follows:
$\begin{eqnarray}{E}_{{\ell },m,n}=\displaystyle \frac{1}{32}{\left((4\,n+3)\tilde{\alpha }+2\,m+1\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}}.\end{eqnarray}$
The corresponding radial wave function will be: $\begin{eqnarray}{{ \mathcal R }}_{{\ell },m,n}={c}_{1}\,{J}_{{\tau }_{{\ell },m}}\left(\displaystyle \frac{\sqrt{2\,{E}_{{\ell },m,n}}}{\tilde{\alpha }}\,r\right),\quad {\tau }_{{\ell },m}=\displaystyle \frac{1}{\tilde{\alpha }}\left(m+\displaystyle \frac{1}{2}\right),\end{eqnarray}$
where c1 is a constant (chosen here to be unity for simplicity) and ∣ℓ∣, m, n = 0, 1, 2,….Then, we can explicitly give some bound states ${{\rm{\Psi }}}_{{\ell },m,n}\left(r,\theta ,\varphi \right)$ and their energy levels Eℓ,m,n for some values of m = 0, 1, 2 and n = 0, 1, 2, 3.
| 1. For m = 0 |
| • | If n = 0, we get: $\begin{eqnarray}{{\rm{\Psi }}}_{{\ell },\mathrm{0,0}}\left(r,\theta ,\varphi \right)={\tilde{C}}_{{\ell },\mathrm{0,0}}\ {r}^{-1/2}{J}_{{\tau }_{{\ell },0}}\left(\displaystyle \frac{\sqrt{2\,{E}_{{\ell },\mathrm{0,0}}}}{\tilde{\alpha }}\,r\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{eqnarray}$ and the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{0,0}}=\displaystyle \frac{1}{32}{\left(1\,+3\tilde{\alpha }\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ with constraints, $\begin{eqnarray}{\lambda }_{{\ell },0}=0,\ {b}_{8}=0,\ {b}_{6}=-\displaystyle \frac{{{\ell }}^{2}}{4}.\end{eqnarray}$ |
| • | If n = 3, |
$\begin{eqnarray}{{\rm{\Psi }}}_{{\ell },\mathrm{0,3}}\left(r,\theta ,\varphi \right)={\tilde{C}}_{{\ell },\mathrm{0,3}}\ {r}^{-1/2}{J}_{{\tau }_{{\ell },0}}\left(\displaystyle \frac{\sqrt{2\,{E}_{{\ell },\mathrm{0,3}}}}{\tilde{\alpha }}\,r\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{eqnarray}$
the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{0,3}}=\displaystyle \frac{1}{32}{\left(1\,+15\tilde{\alpha }\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$
with the same constraints (| 2. For m = 1 |
| • | If n = 1, we have: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{1,1}}\left(r,\theta ,\varphi \right)\\ \,={\tilde{C}}_{{\ell },\mathrm{1,1}}\ {r}^{-1/2}{J}_{{\tau }_{{\ell },1}}\left(\displaystyle \frac{\sqrt{2\,{E}_{{\ell },\mathrm{1,1}}}}{\tilde{\alpha }}\,r\right)\left(\sin \theta +\displaystyle \frac{\sqrt{2}}{2}\right)\,\exp \,\left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{1,1}}=\displaystyle \frac{1}{32}{\left(3\,+7\tilde{\alpha }\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ with constraints, $\begin{eqnarray}{\lambda }_{{\ell },1}=2,\ {b}_{8}=\displaystyle \frac{1}{\sqrt{2}},\ {b}_{6}=-\displaystyle \frac{{{\ell }}^{2}}{4}.\end{eqnarray}$ |
| • | If n = 2 , we get: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{1,2}}\left(r,\theta ,\varphi \right)\\ \,={\tilde{C}}_{{\ell },\mathrm{1,2}}\ {r}^{-1/2}{J}_{{\tau }_{{\ell },1}}\left(\displaystyle \frac{\sqrt{2\,{E}_{{\ell },\mathrm{1,2}}}}{\tilde{\alpha }}\,r\right)\left(\sin \theta -\displaystyle \frac{\sqrt{2}}{2}\right)\,\exp \,\left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ and the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{1,2}}=\displaystyle \frac{1}{32}{\left(3\,+11\tilde{\alpha }\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ with constraints, $\begin{eqnarray}{\lambda }_{\ell ,1}=2,\ {b}_{8}=-\displaystyle \frac{1}{\sqrt{2}},\ {b}_{6}=-\displaystyle \frac{{\ell }^{2}}{4},\end{eqnarray}$ where ${\tau }_{{\ell },1}=\tfrac{3}{2\,\tilde{\alpha }}$ and ∣ℓ∣ = 0, 1, 2, …. |
| 3. For m = 2 |
| • | If n = 2 , we have: $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{{\ell },\mathrm{2,2}}\left(r,\theta ,\varphi \right)\\ \,={\tilde{C}}_{{\ell },\mathrm{2,2}}\ {r}^{-1/2}{J}_{{\tau }_{{\ell },2}}\left(\displaystyle \frac{\sqrt{2\,{E}_{{\ell },\mathrm{2,2}}}}{\tilde{\alpha }}\,r\right)\left({\sin }^{2}\theta -\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4}\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{array}\end{eqnarray}$ the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{2,2}}=\displaystyle \frac{1}{32}{\left(5\,+11\tilde{\alpha }\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ with constraints, $\begin{eqnarray}{\lambda }_{{\ell },2}=6,\ {b}_{8}=-\displaystyle \frac{\sqrt{22}}{2},\ {b}_{6}=-\displaystyle \frac{{{\ell }}^{2}}{4}.\end{eqnarray}$ |
| • | If n = 3 , we obtain: $\begin{eqnarray}{{\rm{\Psi }}}_{{\ell },\mathrm{2,3}}\left(r,\theta ,\varphi \right)={\tilde{C}}_{{\ell },\mathrm{2,3}}\ {r}^{-1/2}{J}_{{\tau }_{{\ell },2}}\left(\displaystyle \frac{\sqrt{2\,{E}_{{\ell },\mathrm{2,3}}}}{\tilde{\alpha }}\,r\right)\left({\sin }^{2}\theta +\displaystyle \frac{\sqrt{22}}{4}\sin \theta +\displaystyle \frac{1}{4}\right)\exp \left({\rm{i}}{\ell }\varphi \right),\end{eqnarray}$ and the associated energy level is, $\begin{eqnarray}{E}_{{\ell },\mathrm{2,2}}=\displaystyle \frac{25}{32}{\left(1\,+3\tilde{\alpha }\right)}^{2}\,\displaystyle \frac{{\pi }^{2}}{{r}_{0}^{2}},\end{eqnarray}$ with constraints, $\begin{eqnarray}{\lambda }_{{\ell },2}=6,\ {b}_{8}=\frac{\sqrt{22}}{2},\ {b}_{6}=-\frac{{{\ell }}^{2}}{4},\end{eqnarray}$ where ${\tau }_{{\ell },2}=\tfrac{5}{2\,\tilde{\alpha }}$ and ∣ℓ∣ = 0, 1, 2,…. |
In the numerical analysis, we have observed that the introduction of a topological defect in the form of a point-like global monopole, characterized by the parameter $\tilde{\alpha }$, significantly impacts the energy levels and the wave functions of quantum particles. This parameter, $\tilde{\alpha }$, associated with the global monopole, notably modifies and reduces the eigenvalue solutions of quantum systems under investigation since it falls within the range of $0\lt \tilde{\alpha }\lt 1$, in contrast to those derived within the background of Minkowski flat space.
We have generated graphs illustrating the energy levels Eℓ,m,n described by equation (71 ) for the scenario where m = 0 in figure 5 and m = 1 in figure 6, considering various values of the topological parameter $\tilde{\alpha }$ and the quantum number n. Similarly, we have depicted graphs showcasing the radial wave function ${{ \mathcal R }}_{{\ell },m,n}$ given by equation (72 ) for m = 0 in figure 7 and m = 1 in figure 8, exploring different values of the topological parameter $\tilde{\alpha }$ and the quantum number n.
Figure 5. Energy levels Eℓ,m,n of equation ( |
Figure 6. Energy levels Eℓ,m,n of equation ( |
Figure 7. Radial wave function ${{ \mathcal R }}_{{\ell },m,n}$ of equation ( |
Figure 8. Radial wave function ${{ \mathcal R }}_{{\ell },m,n}$ of equation ( |
4. Conclusion
In quantum systems, researchers have extensively explored the topological defects stemming from cosmic strings, point-like global monopoles, cosmic dislocations and dispirations, investigating both relativistic and non-relativistic scenarios and scrutinizing eigenvalue solutions. In addition, a plethora of authors have delved into various interacting potential models, including the linear confining potential, Coulomb potential, harmonic oscillator potential, Kratzer potential, generalized Cornell potential, Manning–Rosen potential, Yukawa potential and other non-central potential models. These models find widespread applications in atomic and molecular physics, nuclear physics and solid-state physics. Moreover, external magnetic fields and quantum flux have been incorporated into numerous investigations, leading to analyses of relativistic and non-relativistic eigenvalue solutions obtained through analytical methods and various types of approximations.
In this study, we address the Schrödinger wave equation for the extended ring-shaped potential within the framework of the QES problem. We first consider the influence of a cosmic string and then a point-like global monopole. In both cases, the radial wave equation is solved using special functions, providing the energy eigenvalues of non-relativistic quantum particles under a hard-wall confining condition, where the wave function vanishes within a finite region of space defined by r = r0. The angular-dependent parts in both scenarios are tackled utilizing the Bethe ansatz method, as adopted in [37, 39, 61] and presented their solutions. Subsequently, numerical values for energy levels and the radial wave functions of quantum particles are presented.
Remarkably, our analysis reveals that the energy eigenvalues and radial wave functions are influenced by the topological defects of the cosmic string, characterized by the parameter α, and the global monopole parameter $\tilde{\alpha }$, for different modes represented by the quantum numbers {ℓ, m, n}. Consequently, these eigenvalue solutions undergo modifications due to the presence of the topological defects compared to the results obtained in flat space. The introduction of topological defects in the geometry induces conical singularities along the axis of symmetry, resulting in a background curvature effect. Notably, we observed that the energy eigenvalues of quantum particles increase in the presence of a cosmic string parameter and decrease due to the global monopole parameter compared to the case in flat space. To elucidate the influence of the topological defect parameter on the non-relativistic energy levels and the radial wave function derived in both sections, we have produced several graphs (figures 1–8) showcasing their behavior since the values of the topological parameters and the quantum numbers vary.
Acknowledgments
We thank the anonymous referees for their positive comments, criticism and helpful suggestions, which greatly improved the present study. FA acknowledges the Inter University Centre for Astronomy and Astrophysics (IUCAA), Pune, India, for granting a visiting associateship.
Conflict of interests
The authors declare no conflicts of interest.