The generalized Klein–Gordon oscillator with Coulomb-type potential in (1+2)-dimensions Gürses space–time

Abstract

In this article, we investigate the relativistic quantum dynamics of spin-0 particles in (1+2)-dimensions Gürses space–time Gürses (Class Quantum Grav 11:2585, 1994) with interactions. We solve the generalized Klein–Gordon oscillator subject to Coulomb-type potential in the considered framework, and evaluate the energy eigenvalues and corresponding wave function, in details.

Appendices

Appendix A: (1+2)-dimensions Gürses metric

The (1+2)-dimensions space–time considered by Gürses in [60] (see Eq. (4), notations are same) is given by

$$\begin{aligned} ds^2=-\phi \,dt^2+2\,q\,dt\,d\theta +\frac{-q^2+h^2\,\psi }{\phi }\, d\theta ^2+\frac{1}{\psi }\,dr^2, \end{aligned}$$

(A.1)

where

$$\begin{aligned}&\phi =a_0,\quad \psi =b_0+\frac{b_1}{r^2} +\frac{3\,\lambda _0}{4}\,r^2,\quad h=e_0\,r,\nonumber \\&q=c_0+\frac{e_0\,\mu }{3}\,r^2,\quad \lambda _0=\lambda +\frac{\mu ^2}{27} \end{aligned}$$

(A.2)

and where $a_0,b_0,b_1,c_0,e_0$ are arbitrary constants.

Setting the constants below

$$\begin{aligned} b_1=0,\quad c_0=0,\quad \lambda _0=0 \end{aligned}$$

(A.3)

into the metric (A.1), we arrive the following metric

$$\begin{aligned} ds^2=-a_0\,dt^2+\frac{2\,e_0\,\mu }{3}\,r^2\,dt\,d\theta +\frac{1}{a_0}\,\left( e^{2}_{0}\,b_0\,r^2-\frac{e^{2}_{0}\,\mu ^{2}}{9}\,r^4\right) \, d\theta ^2+\frac{1}{b_0}\,dr^2.\nonumber \\ \end{aligned}$$

(A.4)

Finally choosing the constants

$$\begin{aligned} a_0=1,\quad b_0=1,\quad e_0=1,\quad \Omega =-\frac{\mu }{3} \end{aligned}$$

(A.5)

into the metric (A.4), we get the study space–time (1) presented in this work.

Appendix B: Brief review of the Nikiforov–Uvarov (NU) method

The Nikiforov–Uvarov method is helpful in order to find eigenvalues and eigenfunctions of the Schrödinger like equation, as well as other second-order differential equations of physical interest. According to this method, the eigenfunctions of a second-order differential equation [61]

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

(B.1)

are given by

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

(B.2)

And that the energy eigenvalues equation

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

(B.3)

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

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

(B.4)

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

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

(B.5)

and

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

(B.6)

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

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

(B.7)

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

The energy eigenvalues equation reduces to

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

(B.8)

Few Laguerre polynomial are

$$\begin{aligned}&L^{(\alpha )}_{0} (x)=1\nonumber \\&L^{(\alpha )}_{1} (x)=1+\alpha -x\nonumber \\&L^{(\alpha )}_{2} (x)=\frac{x^2}{2}-(\alpha +2)\,x+\frac{1}{2}\, (\alpha +1)(\alpha +2)\nonumber \\&L^{(\alpha )}_{3} (x)=-\frac{x^3}{6}+\frac{1}{2}\,(\alpha +3)\, x^3-\frac{1}{2}\,(\alpha +2)(\alpha +3)\,x+\frac{1}{2}\,(\alpha +1) (\alpha +2)(\alpha +3).\nonumber \\ \end{aligned}$$

(B.9)

Noted that the simple Laguerre polynomial is the special case $\alpha =0$ of the generalized Laguerre polynomila:

$$\begin{aligned} L^{(0)}_{n} (x)=L_{n} (x). \end{aligned}$$

(B.10)