Eigenvalue spectra of non-relativistic particles confined by AB-flux field with Eckart plus class of Yukawa potential in point-like global monopole

Abstract

In this work, we solve the radial Schrödinger wave equation in three dimensions under the Aharonov–Bohm (AB) flux field with Eckart plus class of Yukawa potential (CYP) in a point-like global monopole (PGM). We determine the approximate eigenvalue solution of the radial equation using the parametric Nikiforov–Uvarov (NU) method and analyze the effects of the topological defect and the magnetic flux field with this potential. Finally, we applied this eigenvalue solution to some potential models, such as Eckart potential, class of Yukawa potential, Hulthen plus Coulomb potential, and Eckart plus Yukawa potential. We show that the presence of the topological defects and the magnetic quantum flux field shifts the eigenvalue solution in comparison to the flat space results.

Appendix: The parametric Nikiforov–Uvarov (NU) method

The Nikiforov–Uvarov method is a helpful technique to calculate the exact and approximate energy eigenvalue and the wave functions of the Schrödinger-like equation and other second-order differential equations of physical interest. According to this method, the wave functions of a second-order differential equation [[63]]

$$\begin{aligned}{} & {} \frac{d^2 \psi (s)}{ds^2}+\frac{(\alpha _1-\alpha _2\,s)}{s\,(1-\alpha _3\,s)}\frac{d \psi (s)}{ds}\nonumber \\{} & {} \quad +\frac{(-\xi _1\,s^2+\xi _2\,s-\xi _3)}{s^2\,(1-\alpha _3\,s)^2}\psi (s)=0 \end{aligned}$$

(A.1)

are given by

$$\begin{aligned}{} & {} \psi (s)\nonumber \\{} & {} \quad =s^{\alpha _{12}}(1-\alpha _3 s)^{-\alpha _{12}-\frac{\alpha _{13}}{\alpha _3}}P^{(\alpha _{10}-1,\frac{\alpha _{11}}{\alpha _3}-\alpha _{10}-1)}_{n}(1-2\alpha _3 s). \end{aligned}$$

(A.2)

And that the energy eigenvalues equation

$$\begin{aligned}{} & {} \alpha _2\,n-(2\,n+1)\,\alpha _5+(2\,n+1)\,(\sqrt{\alpha _9}+\alpha _3\,\sqrt{\alpha _8})\nonumber \\{} & {} \quad +n\,(n-1)\,\alpha _3+\alpha _7+2\,\alpha _3\,\alpha _8+2\,\sqrt{\alpha _8\,\alpha _9}=0. \end{aligned}$$

(A.3)

The parameters $\alpha _4,\ldots ,\alpha _{13}$ are obatined from the six parameters $\alpha _1,\ldots ,\alpha _3$ and $\xi _1,\ldots ,\xi _3$ as follows:

$$\begin{aligned}{} & {} \alpha _4=\frac{1}{2}\,(1-\alpha _1),\quad \alpha _5=\frac{1}{2}\,(\alpha _2-2\,\alpha _3),\quad \alpha _6=\alpha ^2_{5}+\xi _1,\nonumber \\{} & {} \quad \alpha _7=2\,\alpha _4\,\alpha _{5}-\xi _2,\quad \alpha _8=\alpha ^2_{4}+\xi _3,\nonumber \\{} & {} \alpha _9=\alpha _6+\alpha _3\,\alpha _7+\alpha ^{2}_3\,\alpha _8,\quad \alpha _{10}=\alpha _1+2\,\alpha _4+2\,\sqrt{\alpha _8},\nonumber \\{} & {} \quad \alpha _{11}=\alpha _2-2\,\alpha _5+2\,(\sqrt{\alpha _9}+\alpha _3\,\sqrt{\alpha _8}),\nonumber \\{} & {} \quad \alpha _{12}=\alpha _4+\sqrt{\alpha _8},\quad \alpha _{13}=\alpha _5-(\sqrt{\alpha _9}+\alpha _3\,\sqrt{\alpha _8}). \end{aligned}$$

(A.4)

A special case where $\alpha _3=0$, we find

$$\begin{aligned} \lim _{\alpha _3\rightarrow 0} P^{(\alpha _{10}-1,\frac{\alpha _{11}}{\alpha _3}-\alpha _{10}-1)}_{n}\,(1-2\,\alpha _3\,s)=L^{\alpha _{10}-1}_{n} (\alpha _{11}\,s), \end{aligned}$$

(A.5)

and

$$\begin{aligned} \lim _{\alpha _3\rightarrow 0} (1-\alpha _3\,s)^{-\alpha _{12}-\frac{\alpha _{13}}{\alpha _3}}=e^{\alpha _{13}\,s}. \end{aligned}$$

(A.6)

Therefore the wave-function from (A.2) becomes

$$\begin{aligned} \psi (s)=s^{\alpha _{12}}\,e^{\alpha _{13}\,s}\,L^{\alpha _{10}-1}_{n} (\alpha _{11}\,s), \end{aligned}$$

(A.7)

where $L^{(\beta )}_{n} (x)$ denotes the generalized Laguerre polynomial.

The energy eigenvalues equation reduces to

$$\begin{aligned}{} & {} n\,\alpha _2-(2\,n+1)\,\alpha _5+(2\,n+1)\,\sqrt{\alpha _9}+\alpha _7\nonumber \\{} & {} \quad +2\,\sqrt{\alpha _8\,\alpha _9}=0. \end{aligned}$$

(A.8)