Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class of Multiparameter Exponential-Type Potentials

Abstract

By using the exactly-solvable Schrödinger equation for a class of multi-parameter exponential-type potential, the analytical bound state solutions of the Klein-Gordon equation are presented. The proposal is based on the fact that the Klein-Gordon equation can be reduced to a Schrödinger-type equation when the Lorentz-scalar and vector potential are equal. The proposal has the advantage of avoiding the use of a specialized method to solve the Klein-Gordon equation for a specific exponential potential due that it can be derived by means of an appropriate choice of the involved parameters. For this, to show the usefulness of the method, the relativistic treatment of spinless particles subject to some already published exponential potentials are directly deduced and given as examples. So, beyond the particular cases considered in this work, this approach can be used to solve the Klein-Gordon equation for new exponential-type potentials having hypergeometric eigenfunctions. Also, it can be easily adapted to other approximations of the centrifugal term different to the Green-Aldrich used in this work.

1. Introduction

At high energy levels, the study of physical phenomena is carried out by means of equations invariant under Lorentz transformations. That is, it requires relativistic wave equations that may be used as a starting point to evaluate the spin-orbit interactions and relativistic effective core potential in the Schrödinger Hamiltonian. The energy levels from these calculations are aimed to find the positions of experimental spectral lines and to predict lines not heretofore observed in the systems under consideration. To that purpose, the Dirac and Klein-Gordon equations are used for the dynamic description of particles with and without spin, respectively. For that, the solutions of these equations have been an important field of research by employing several methods as well as different physical potential models; usually a solution method for a specific potential. In this regard, exponential-type potentials are significant in the study of various physical systems, particularly for modeling diatomic molecules. Within the different exponential potential models, stand out the proposals of Hulthén, Eckart, Manning-Rosen, Rosen-Morse, Deng-Fan, Hyllerass, etc., as well as mixed models with two or more of the above potential models. These latter, have been developed with the aim to solve, as particular cases, the specific potentials that are involved. In any case, the common feature of any exponential-type potential is that wave functions are of hypergeometric-type. For this reason, in the quantum mechanics treatment of this kind of potentials, the method that is most often used to find the bound states solutions is the Nikiforov-Uvarov method [1], which is based on solving a hypergeometric-type differential equation (DE) by means of special orthogonal functions. Albeit, other procedures such as Asymptotic Iteration [2], Supersymmetric Quantum Mechanics [3], He’s Variational iteration [4], large-N solutions [5] or Quantization-rule [6], among many other methods, have been also employed in both non-relativistic and relativistic studies; obviously, including numerical solutions [7]. In the relativistic studies of spinless particles, it is well known that the Klein-Gordon equation [8, 9] can always be reduced to a Schrödinger-type equation when the Lorentz-scalar and vector potential are equal [10]. This fact is used in the present research, devoted to obtaining approximate bound state energy eigenvalues and the corresponding eigenfunctions of the Klein-Gordon equation for exponential-type potentials. The method is based on a direct approach applied to the exactly solvable Schrödinger equation with hypergeometric solutions for exponential-type potential [11], which is given in Section 2. With this result, the corresponding analytical bound state solutions of the Klein-Gordon equation are found in the frame of the Green and Aldrich approximation to the centrifugal term [12] as shown in Section 4. Advantageously, according to the method, several specific potentials are derived as particular cases from the proposal such as those given in Section 5.

2. Direct approach to the exactly solvable Schrödinger equation with hypergeometric solutions

For finding exactly-solvable quantum exponential-type potentials, the Schrödinger equation must be transformed into a hypergeometric differential equation. To do so, let us consider the Schrödinger equation (${\mathit{\text{ℏ}}}^{\mathbf{2}}\mathit{=}\mathbf{2}\mathit{m}\mathit{=}\mathbf{1}$)

$$\frac{-d^2\psi \left(r\right)}{d{r}^2}+V\left(r\right)\psi \left(r\right)=\mathit{E\psi }\left(r\right)$$E1

such that, after using the transformation

$$\psi \left(r\right)=e^{-\mathit{\alpha r}}u\left(r\right)$$E2

it is written as

$$\frac{-d^{2}u\left(r\right)}{d{r}^2}+2\alpha \frac{\mathit{du}\left(r\right)}{\mathit{dr}}+\left[V\left(r\right)-\left(E+{\alpha }^2\right)\right]u\left(r\right)=0$$E3

With the aim of relating the above equation with a hypergeometric DE, we use the coordinate transformation

$$x=q{e}^{-r/k},k>0,q\ne 0$$E4

such that Eq. (3) is written as

$$x\left(1-x\right)\frac{d^{2}u\left(x\right)}{d{x}^2}-\left(1+2\mathit{\alpha k}\right)\left(1-x\right)\frac{\mathit{du}\left(x\right)}{\mathit{dx}}-k^2\left(\frac{1-x}{x}\right)\left[V\left(x\right)-\left(E+{\alpha }^2\right)\right]u\left(x\right)=0.$$E5

Then, the similarity transformation

$$u\left(x\right){\left(1-x\right)}^{d}F\left(x\right)$$E6

with d being a real parameter, gives rise to

$$x\left(1-x\right)F^{\prime \prime }\left(x\right)+\left[\left(1+2\mathit{\alpha k}\right)-\left(1+2\mathit{\alpha k}+2d\right)x\right]F^{\prime }\left(x\right)+R\left(x\right)F\left(x\right)=0$$E7

where

$$R\left(x\right)=\frac{\mathit{xd}\left(d-1\right)}{1-x}-d\left(1+2\mathit{\alpha k}\right)-k^2\left(\frac{1-x}{x}\right)\left[V\left(x\right)-\left(E+{\alpha }^2\right)\right];$$E8

Hence, Eq. (7) can be compared with a hypergeometric DE

$$x\left(1-x\right)y^{\prime \prime }\left(x\right)+\left[c-\left(1+a+b\right)x\right]y^{\prime }\left(x\right)-\mathit{aby}\left(x\right)=0$$E9

provided that

$$1+2\mathit{\alpha k}=c,2\left(\mathit{k\alpha }+d\right)=a+b\mathrm{,}E=-{\alpha }^2$$E10

and

$$R\left(x\right)=-\mathit{ab}\wedge F\left(x\right)=y\left(x\right)={{}_{2}F}_1\left(a,b;c:x\right),$$E11

where ${{}_{2}F}_1$ is the hypergeometric function.

So, from $R=\mathit{ab}$ and $E=-{\alpha }^2$ in Eq. (8), it is possible to identify the potential

$$V\left(x\right)=\frac{1}{k^2}\left[\frac{\left(\mathit{ab}-\left(1+2\mathit{k\alpha }\right)d\right)x}{1-x}+\frac{d\left(d-1\right)x^2}{{\left(1-x\right)}^2}\right]$$E12

which, by using Eqs. (4) and (10), can be written as

$$V\left(r\right)=\frac{\left[4\mathit{ab}-2c\left(a+b+1-c\right)\right]q{e}^{-r/k}+\left[{\left(a-b\right)}^2-{\left(c-1\right)}^2\right]q^2{e}^{-2r/k}}{4k^2{\left(1-q{e}^{-r/k}\right)}^2}$$E13

with eigenfunction given from Eqs. (2) and (6) by

$$\psi \left(r\right)=e^{-\mathit{\alpha r}}{\left(1-q{e}^{-r/k}\right)}^d{{}_{2}F}_1\left(a,b;c:q{e}^{-r/k}\right)$$E14

and eigenvalue

$$E=-{\alpha }^2=-{\left(\frac{c-1}{2k}\right)}^2.$$E15

At this point, it is convenient to introduce the new parameters $A,B$, and $C$ such that $4k^2\left(A+B\right)=4\mathit{ab}-2c\left(a+b+1-c\right)$ and $4k^2\left(C-A\right)={\left(a-b\right)}^2-{\left(c-1\right)}^2$ in order to rewrite the potential as a multi-parameter exponential-type potential

$$V\left(r\right)=\frac{\mathit{Aq}{e}^{-r/k}}{1-q{e}^{-r/k}}+\frac{\mathit{Bq}{e}^{-r/k}}{{\left(1-q{e}^{-r/k}\right)}^2}+\frac{C{q}^2{e}^{-2r/k}}{{\left(1-q{e}^{-r/k}\right)}^2}$$E16

By solving the condition $\frac{d}{\mathit{dr}}V\left(r\right)=0$, there will be a minimum value for the potential with sufficient depth for the existence of bound states, namely

$$V\left(r_{\mathit{min}}\right)=\frac{-{\left(A+B\right)}^2}{4\left(B+C\right)}$$

$$\mathit{\text{with}}{r}_{\mathit{min}}=\mathit{kln}\left(\frac{q\left(A-B-2C\right)}{A+B}\right),$$E17

provided that

$$A+B<0\le B+C$$E18

which ensures that $V\left(r\right)$ is an attractive potential with an infinite wall at its singular point $r_s=\mathit{kln}\left(q\right)$.

Regarding the wave function, in order to have a node at $r_s$ it is necessary to apply the condition $\psi \left(r_s\right)=0$, which is achieved if $d>0$. Furthermore, by combining the identities $4k^2\left(A+B\right)$ and $4k^2\left(C-A\right)$ given above, we have

$$b=\frac{\left(h+1\right)\left(h-a\right)-2k^2\left(A+B\right)}{h+1-2a}\mathrm{,}c=\frac{2a\left(h-a\right)-2k^2\left(A+B\right)}{h+1-2a}$$E19

such that

$$c=a+b-\mathit{\text{h}}\mathit{\text{with}}\mathit{\text{h}}=\sqrt{1+4k^2\left(B+C\right)}$$E20

besides, if the parameter $a=-n,n=0,1,2,3\dots .$, the hypergeometric function appearing in the eigenfunction given by Eq. (14) becomes a polynomial of $n-\mathit{th}$ degree in the variable $q{e}^{-r/k}$. Additionally, the condition $\psi \left(r\right)⟶$ 0 when $r⟶\infty$ implies that $\alpha =\frac{c-1}{k}>0$, from which, the number of states is

$$0\le n<k\sqrt{C-A}-\frac{h+1}{2}$$E21

In short, the above equations lead to a well define wave function for a legitimate Schrödinger equation with potential $V\left(r\right)$. Likewise, by using Eqs. (19) and (20) in Eq. (15) the energy spectrum will be

$$E_n=\frac{-1}{4k^2}{\left[\frac{{\left(n+\frac{\left(h+1\right)}{2}\right)}^2+k^2\left(A-C\right)}{n+\frac{\left(h+1\right)}{2}}\right]}^2$$E22

with wave-functions

$${\psi }_n={\left(e^{-r/k}\right)}^{\frac{c-1}{2}}{\left(1-q{e}^{-r/k}\right)}^{\frac{h+1}{2}}{{}_{2}F}_1\left(-n,b;c:q{e}^{-r/k}\right)$$E23

3. Klein-Gordon equation in arbitrary dimensions

For a spinless particle with energy $E_{\mathit{nl}}$ and mass $M$, the D-dimensional Klein-Gordon (KG) equation is given ($\hslash =c=1$) by [13].

$${-\nabla }_D^2{\psi }_{\mathit{nlm}}\left(r,\Omega \right)+\left\{{\left[M+S\left(r\right)\right]}^2-{\left[E_{\mathit{nl}}+V\left(r\right)\right]}^2\right\}{\psi }_{\mathit{nlm}}\left(r,\Omega \right)=0$$E24

where $V\left(r\right)$ and $S\left(r\right)$ are respectively the Lorentz vector and the scalar interaction potentials. The D-dimensional Laplacian operator in the space $\left(r,\Omega \right)=\left(r,{\theta }_1,{\theta }_2,{\theta }_3,\dots {\theta }_{D-2},\varphi \right)$ is defined ase

$${\nabla }_D^2=r^{1-D}\frac{\partial }{\partial r}\left[r^{D-1}\frac{\partial }{\partial r}\right]-\frac{{\Lambda }^2\left(\Omega \right)}{r^2}$$E25

with $\Lambda \left(\Omega \right)$ the angular momentum operator. Hence, the functione

$${\psi }_{\mathit{nlm}}\left(r,\Omega \right)=R_{\mathit{nl}}\left(r\right)Y_l^m\left(\Omega \right)$$E26

leads to the radial part of Eq. (24)

$$\frac{-d^2{R}_{\mathit{nl}}\left(r\right)}{d{r}^2}-\frac{D-1}{r}\frac{d{R}_{\mathit{nl}}\left(r\right)}{\mathit{dr}}+\frac{l_D\left(l_D+1\right)}{r^2}{R}_{\mathit{nl}}\left(r\right)$$

$$+\left\{{\left[M+S\left(r\right)\right]}^2-{\left[E_{\mathit{nl}}+V\left(r\right)\right]}^2\right\}{R}_{\mathit{nl}}\left(r\right)=E{R}_{\mathit{nl}}\left(r\right)$$E27

where we have used $l_D=\sqrt{\frac{1}{4}+l\left(l+D-2\right)}-\frac{1}{2}$ such that

$${\Lambda }^2\left(\Omega \right)Y_l^m\left(\Omega \right)=l_D\left(l_D+1\right)Y_l^m\left(\Omega \right)$$E28

Likewise, with $R_{\mathit{nl}}\left(r\right)=r^{\frac{-D-1}{2}}{\psi }_{\mathit{nL}}\left(r\right)$Eq. (27) becomes

$$\frac{-d^2{\psi }_{\mathit{nL}}\left(r\right)}{d{r}^2}+\left\{{\left[M+S\left(r\right)\right]}^2-{\left[E_{\mathit{nL}}+V\left(r\right)\right]}^2+\frac{L_s}{r^2}\right\}{\psi }_{\mathit{nL}}\left(r\right)=E_{\mathit{nL}}{\psi }_{\mathit{nL}}\left(r\right)$$E29

where $L_s=L\left(L+1\right)$, $L=l_D+\frac{D-3}{2}$ such that the case $D=3$ implies $L=l_D=l.$

At this point, as already mentioned, the D-dimensional KG equation given in Eq. (29) can be reduced to a Schrödinger-like equation, provided that the Lorentz vector and scalar potential are equal [10]. In fact, if $V\left(r\right)=S\left(r\right)$ the corresponding KG equation is given by

$$\frac{-d^2{\psi }_{\mathit{nL}}\left(r\right)}{d{r}^2}+\left\{\left(E_{\mathit{nL}}^2-M^2\right)-2\left(E_{\mathit{nL}}+M\right)V\left(r\right)+\frac{L_s}{r^2}\right\}{\psi }_{\mathit{nL}}\left(r\right)=0$$E30

Different methods of the solution have been applied for solving the above equation with many models of interaction potentials; see for example Ikhdair [14] and references therein. Hence, to provide a unified treatment to the bound states solution of the KG equation for equal vector and scalar exponential-type potentials, in the next paragraph the results of Section 2 are extended to consider the D-dimensional case. As we will see, this is done by a simple redefinition of the parameters that appear in $V\left(r\right)$ as defined in Eq. (16).

4. Klein-Gordon equation for exponential-type potentials in arbitrary dimensions

To deal with the 𝓁-state approximate solutions for the D-dimensional KG equation with the multi-parameter exponential-type potential given in Eq. (16) we define $q=1$ and

$$A=A+\frac{\alpha L_s}{k^2},B=B+\frac{\beta L_s}{k^2},C=C+\frac{\gamma L_s}{k^2}$$E31

such that

$$V\left(r\right)=\frac{A{e}^{-r/k}}{1-e^{-r/k}}+\frac{B{e}^{-r/k}}{{\left(1-e^{-r/k}\right)}^2}+\frac{C{e}^{-2r/k}}{{\left(1-e^{-r/k}\right)}^2}+L_s{T}_c$$E32

with

$$T_c=\frac{\left(\alpha +\beta \right)e^{-r/k}+\left(\gamma -\alpha \right)e^{-2r/k}}{k^2{\left(1-e^{-r/k}\right)}^2}$$E33

where, according to the values of the parameters α, β, and γ, the function $T_c$ would be approximate to the centrifugal term. Consequently, this method accepts different approximation schemes, such as recently shown in [15]. For example, if we consider the case $\alpha =\gamma =0,$ and $\beta =1$, it leads to the standard Green-Aldrich approximation [12]

$$T_c=\frac{e^{-r/k}}{k^2{\left(1-e^{-r/k}\right)}^2}\approx \frac{1}{r^2}$$E34

however, if we add the constant $c_{0}L/k^2$ in both sides of the Eq. (32) one has

$$V\left(r\right)+\frac{c_{0}L}{k^2}=\frac{A{e}^{-r/k}}{1-e^{-r/k}}+\frac{B{e}^{-r/k}}{{\left(1-e^{-r/k}\right)}^2}+\frac{\mathit{Cq}{e}^{-2r/k}}{{\left(1-e^{-r/k}\right)}^2}+L_s\left[T_c+\frac{c_0}{k^2}\right]$$E35

which means that any improvement to the centrifugal term through an additive constant will be reflected as an additional term in the energy spectrum. In fact, the improved Green-Aldrich approximation to the centrifugal term is achieved when $\alpha =\gamma =0$, $\beta =1$ and $c_0=1/12$, that is [16]

$$T_c+\frac{c_0}{k^2}=\frac{e^{-r/k}}{k^2{\left(1-e^{-r/k}\right)}^2}+\frac{1}{12k^2}\approx \frac{1}{r^2}$$E36

Another typical improved approximation used to $\frac{1}{r^2}$ is when $\alpha =C_1$; $\beta =0$, and $\gamma =C_2$ and $c_0=C_0$ where the parameters $C_1$, $C_2$, and $C_0$ are adjustable parameters [17], leading to

$$T_c+\frac{c_0}{k^2}=\frac{1}{k^2}\left[C_0+\frac{C_1{e}^{-r/k}}{1-e^{-r/k}}+\frac{C_2{e}^{-2r/k}}{{\left(1-e^{-r/k}\right)}^2}\right]\approx \frac{1}{r^2}$$E37

However, for the sake of simplicity, we will use the standard Green-Aldrich approximation, $C_0=0$, such that

$$V\left(r\right)=V\left(r\right)+\frac{L_s}{r^2}$$E38

with

$$V\left(r\right)=\frac{A{e}^{-r/k}}{1-e^{-r/k}}+\frac{B{e}^{-r/k}}{{\left(1-e^{-r/k}\right)}^2}+\frac{C{e}^{-2r/k}}{{\left(1-e^{-r/k}\right)}^2}$$E39

Since the solutions of Eq. (1) with potential $V\left(r\right)$ are given by Eqs. (22) and (23), the energy spectrum and the eigenfunctions will be

$$E_n=\frac{-1}{4k^2}{\left[\frac{{\left(n+\frac{\left(h_L+1\right)}{2}\right)}^2+k^2\left(A-C\right)}{n+\frac{\left(h_L+1\right)}{2}}\right]}^2$$E40

and

$${\psi }_{\mathit{nL}}\left(r\right)={\left(e^{-r/k}\right)}^{\frac{C_L-1}{2}}{\left(1-e^{-r/k}\right)}^{\frac{h_L+1}{2}}{{}_{2}F}_1\left(-n,b_L;c_L:e^{-r/k}\right)$$E41

where the new parameters defined in Eq. (31) have been used. Besides, according with Eqs. (19) and (20)

$$h_L=\sqrt{{\left(2L+1\right)}^2+4k^2\left(B+C\right)},$$E42

$$b_L=\frac{\left(h_L+1\right)\left(h_L+n\right)-2k^2\left(A+B\right)-2L_s}{2n+h_L+1}$$E43

and

$$c_L=\frac{-2n\left(h_L+n\right)-2k^2\left(A+B\right)-2L_s}{2n+h_L+1}$$E44

Furthermore, the number of states will be determined by Eq. (21) as

$$0\le n<k\sqrt{C-A}-\frac{h_L+1}{2}$$E45

Likewise, in accordance with Eq. (17)

$$V\left(r_{\mathit{min}}\right)=\frac{-{\left(k^2\left(A+B\right)+L_s\right)}^2}{4k^2\left(k^2\left(B+C\right)+L_s\right)}$$E46

with

$$r_{\mathit{min}}=\mathit{kln}\left(\frac{k^2\left(A-B-2C\right)-L_s}{k^2\left(A+B\right)+L_s}\right),$$E47

such that

$$k^2\left(A+B\right)<L_s\le k^2\left(B+C\right)$$E48

or more explicitly

$$k^2\left(A+B\right)+l\left(l+1\right)<\frac{\left(3-D\right)\left(4l+D-1\right)}{4}\le k^2\left(B+C\right)+l\left(l+1\right)$$E49

where all possible values of $l=0,1,2,3,\dots l_{\mathit{max}}$ fulfill the above inequality.

With these elements, the KG equation in arbitrary dimensions given in Eq. (30), for $S\left(r\right)=V\left(r\right)=V\left(r\right)$; becomes a Schrödinger-type equation

$$\frac{-d^2{\psi }_{\mathit{nL}}\left(r\right)}{d{r}^2}+\left\{∆\mathit{EV}\left(r\right)+\frac{L_s}{r^2}\right\}{\psi }_{\mathit{nL}}\left(r\right)={\stackrel{\sim }{E}}_{\mathit{nL}}{\psi }_{\mathit{nL}}\left(r\right)$$E50

with $∆E=2\left(E_{\mathit{nL}}+M\right)$ and ${\stackrel{\sim }{E}}_{\mathit{nL}}=E_{\mathit{nL}}^2-M^2$ Hence, within the frame of the standard Green-Aldrich approximation, directly from Eqs. (40) and (41), the energy spectrum and wave function are respectively

$${\stackrel{\sim }{E}}_{\mathit{nL}}=\frac{-1}{16k^2}{\left[\frac{{\left(2n+h_L+1\right)}^2+4k^2∆E\left(A-C\right)}{2n+h_L+1}\right]}^2$$E51

$${\psi }_{\mathit{nL}}\left(r\right)={\left(e^{-r/k}\right)}^{\frac{C_L-1}{2}}{\left(1-e^{-r/k}\right)}^{\frac{h_L+1}{2}}{{}_{2}F}_1\left(-n,b_L;c_L:e^{-r/k}\right)$$E52

where

$$h_L=\sqrt{{\left(2L+1\right)}^2+4k^2∆E\left(B+C\right)},$$E53

$$b_L=\frac{\left(h_L+1\right)\left(h_L+n\right)-2k^2∆E\left(A+B\right)-2L_s}{2n+h_L+1}$$E54

and

$$c_L=\frac{-2n\left(h_L+n\right)-2k^2∆E\left(A+B\right)-2L_s}{2n+h_L+1}$$E55

The usefulness of our alternative approach for the calculation of bound state solutions of the D-dimensional KG equation with exponential-type potentials is exemplified in the next section.

5. Applications

The choice of particular values for the parameters $A$, $B$, and $C$ appearing in the multiparameter exponential-type potential of Eq. (39), leads to the solutions of the KG equation in arbitrary dimensions for specific potentials. So, without being exhaustive, at the following we are going to consider only some well-known special cases, it being understood the existence of many others that can be treated in a similar way.

5.1 The Eckart+Hultén potential

If we assume the parameters $A=-\left(V_2+V_3\right);B=4V_1;C=0$ and $k={\left(2a\right)}^{-1}$ the potential $V\left(r\right)$ in Eq. (39) will be

$$V\left(r\right)=\frac{-\left(V_2+V_3\right)e^{-2\mathit{\alpha r}}}{1-e^{-2\mathit{\alpha r}}}+\frac{4V_1{e}^{-2\mathit{\alpha r}}}{{\left(1-e^{-2\mathit{\alpha r}}\right)}^2}$$E56

such that the Eckart-type potential which also includes the Hulthén potential is written as

$$V_{\mathit{EH}}=V\left(r\right)-V_2=\frac{-V_2}{1-e^{-2\mathit{\alpha r}}}-\frac{V_3{e}^{-2\mathit{\alpha r}}}{1-e^{-2\mathit{\alpha r}}}+\frac{4V_1{e}^{-2\mathit{\alpha r}}}{{\left(1-e^{-2\mathit{\alpha r}}\right)}^2}+\frac{L_s}{r^2}$$E57

Hence, from Eq. (51) this potential has an energy spectrum given by

$${\stackrel{\sim }{E}}_{\mathit{nL}}^{\mathit{EH}}=\frac{-{\alpha }^2}{4}{\left[\frac{{\left(2n+h_L+1\right)}^2+{\alpha }^{-2}∆E\left(V_2+V_3\right)}{2n+h_L+1}\right]}^2-∆E{V}_2$$E58

which agrees with the transcendental Eq. (38) of the reference [14], after considering some algebraic steps on it and the displacement $-V_2$ in the potential $V\left(r\right)$. At this point, we want to notice that the term $∆E{V}_2$ in above equation appears from $V\left(r\right)=V_{\mathit{EH}}+V_2$ given in Eq. (57). That is,

$$\frac{-d^2{\psi }_{\mathit{nL}}\left(r\right)}{d{r}^2}+\left[∆E\left(V_{\mathit{EH}}+V_2\right)+\frac{L_s}{r^2}\right]{\psi }_{\mathit{nL}}\left(r\right)={\stackrel{\sim }{E}}_{\mathit{nL}}{\psi }_{\mathit{nL}}\left(r\right)$$E59

Implies

$$\frac{-d^2{\psi }_{\mathit{nL}}\left(r\right)}{d{r}^2}+\left[∆E{V}_{\mathit{EH}}+\frac{L_s}{r^2}\right]{\psi }_{\mathit{nL}}\left(r\right)=\left({\stackrel{\sim }{E}}_{\mathit{nL}}-∆E{V}_2\right){\psi }_{\mathit{nL}}\left(r\right)$$E60

Likewise from Eq. (52), the corresponding wave functions are

$${\psi }_{\mathit{nL}}\left(r\right)={\left(e^{-2\mathit{\alpha r}}\right)}^{\frac{C_{\mathit{EH}\left(L\right)}-1}{2}}{\left(1-e^{-2\mathit{\alpha r}}\right)}^{\frac{h_{\mathit{EH}\left(L\right)}+1}{2}}{{}_{2}F}_1\left(-n,b_{\mathit{EH}\left(L\right)};c_{\mathit{EH}\left(L\right)};e^{-2\mathit{\alpha r}}\right)$$E61

where

$$h_L=\sqrt{{\left(2L+1\right)}^2+{\alpha }^{-2}∆E{V}_1},$$E62

$$b_{\mathit{EH}\left(L\right)}=\left(h_{\mathit{EH}\left(L\right)}+1\right)\left(h_{\mathit{EH}\left(L\right)}+n\right)-$$E63

and

$$c_{\mathit{EH}\left(L\right)}=-2n\left(h_{\mathit{EH}\left(L\right)}+n\right)-2$$E64

5.2 The standard Hultén potential

In this case, for the potential $V\left(r\right)$ given in Eq. (39), one can apply the selection

$$A=-\mathit{Z\alpha },B=C=0,k={\alpha }^{-1}$$E65

such that the Hultén potential is

$$V_H=\frac{-\mathit{Z\alpha }{e}^{-\mathit{\alpha r}}}{1-e^{-\mathit{\alpha r}}}+\frac{L_s}{r^2}$$E66

Similarly to the above case, from Eq. (51), this potential has an energy spectrum

$${\stackrel{\sim }{E}}_{\mathit{nL}}^H=\frac{-{\alpha }^2}{4}{\left[n+L+1-\frac{\mathit{Z\alpha }∆E}{n+L+1}\right]}^2$$E67

that agrees with Eq. (47) of the reference [14] under the identification of their $\nu =\left(D+2l-1\right)/2$ with our $L=\nu -1$ i.e. $\nu =L+1$. Additionally, our ${\stackrel{\sim }{E}}_{\mathit{nL}}^H$ result coincides with that of Saad [18] when the parameters $V_0=S_0$ and $q=1$ are used. Similarly, from Eq. (52), the corresponding wavefunctions will be

$${\psi }_{\mathit{nL}}^H\left(r\right)={\left(e^{-\mathit{\alpha r}}\right)}^{\frac{C_{H\left(L\right)}-1}{2}}{\left(1-e^{-\mathit{\alpha r}}\right)}^{L+1}{{}_{2}F}_1\left(-n,b_{H\left(L\right)};c_{H\left(L\right)};e^{-\mathit{\alpha r}}\right)$$E68

where, according to Eqs. (54) and (55), the $b_L$ and $b_L$ parameters are now

$$b_{H\left(L\right)}=\frac{\left(L+1\right)\left(2L+n+1\right)+Z{\alpha }^{-1}-L_s}{n+L+1}$$E69

and

$$c_{H\left(L\right)}=\frac{-n\left(2L+n+1\right)+Z{\alpha }^{-1}-L_s}{n+L+1}$$E70

with $h_{H\left(L\right)}=2L+1$.

5.3 The standard Eckart potential

To obtain this potential model, one selects

$$A=-V_1,B=V_2,C=0;k=b$$E71

such that

$$V_E=\frac{-V_1{e}^{-r/b}}{1-e^{-r/b}}+\frac{V_2{e}^{-r/b}}{{\left(1-e^{-r/b}\right)}^2}+\frac{L_s}{r^2}$$E72

corresponds to the Eckart potential. So, from Eq. (51), its corresponding energy spectrum results in

$${\stackrel{\sim }{E}}_{\mathit{nL}}^E=\frac{-1}{16b^2}{\left[\frac{{\left(2n+h_{E\left(L\right)}+1\right)}^2+4b^2{V}_1∆E}{2n+h_{E\left(L\right)}+1}\right]}^2$$E73

where

$$h_{E\left(L\right)}=\sqrt{{\left(2L+1\right)}^2+4b^2∆E{V}_2},$$E74

and the eigenfunctions are

$${\psi }_{\mathit{nL}}^E\left(r\right)={\left(e^{-\mathit{\alpha r}}\right)}^{\frac{C_{E\left(L\right)}-1}{2}}{\left(1-e^{-\mathit{\alpha r}}\right)}^{\frac{h_{E\left(L\right)}+1}{2}}{{}_{2}F}_1\left(-n,b_{E\left(L\right)};c_{E\left(L\right)};e^{-\mathit{\alpha r}}\right)$$E75

$$b_{\mathit{EH}\left(L\right)}=\frac{\left(h_{E\left(L\right)}+1\right)\left(h_{E\left(L\right)}+n\right)-2b^2∆E\left(V_2-V_1\right)-2L_s}{2n+h_{E\left(L\right)}+1}$$E76

and

$$c_{\mathit{EH}\left(L\right)}=\frac{-2n\left(h_{E\left(L\right)}+n\right)-2b^2∆E\left(V_2-V_1\right)-2L_s}{2n+h_{E\left(L\right)}+1}$$E77

It is worth mentioning that in the particular case $D=3$, the energy spectrum given in Eq. (73) coincides with the results of Akpan et al. [17] by assuming the standard Green and Aldrich approximation ($C_0=0$; $C_1=C_2=1$).

5.4 The Manning-Rosen potential

Let us consider now the parameters

$$A=-V_0/b^2,B=0,C=\frac{\alpha \left(\alpha -1\right)}{b^2};k=b$$E78

from which, one has the Manning-Rosen potential

$$V_{\mathit{MR}}=\frac{1}{b^2}\left(\frac{\alpha \left(\alpha -1\right)e^{-2r/b}}{{\left(1-e^{-r/b}\right)}^2}-\frac{V_0{e}^{-r/b}}{1-e^{-r/b}}\right)+\frac{L_s}{r^2}$$E79

with energy spectrum

$${\stackrel{\sim }{E}}_{\mathit{nL}}^{\mathit{MR}}=\frac{-1}{16b^2}{\left[\frac{{\left(2n+h_{\mathit{MR}\left(L\right)}+1\right)}^2+4∆E\left(V_0+\alpha \left(\alpha -1\right)\right)}{2n+h_{\mathit{MR}\left(L\right)}+1}\right]}^2$$E80

where

$$h_{\mathit{MR}\left(L\right)}=\sqrt{{\left(2L+1\right)}^2+4∆\mathit{E\alpha }\left(\alpha -1\right)},$$E81

Besides, the eigenfunctions are

$${\psi }_{\mathit{nL}}^{\mathit{MR}}\left(r\right)={\left(e^{-r/b}\right)}^{\frac{C_{\mathit{MR}\left(L\right)}-1}{2}}{\left(1-e^{-r/b}\right)}^{\frac{h_{\mathit{MR}\left(L\right)}+1}{2}}{{}_{2}F}_1\left(-n,b_{\mathit{MR}\left(L\right)};c_{\mathit{MR}\left(L\right)};e^{-r/b}\right)$$E82

$$b_{\mathit{MR}\left(L\right)}=\frac{\left(h_{\mathit{MR}\left(L\right)}+1\right)\left(h_{\mathit{MR}\left(L\right)}+n\right)+2∆E{V}_0-2L_s}{2n+h_{\mathit{MR}\left(L\right)}+1}$$E83

and

$$c_{\mathit{MR}\left(L\right)}=\frac{-2n\left(h_{\mathit{MR}\left(L\right)}+n\right)+2∆E{V}_0-2L_s}{2n+h_{\mathit{MR}\left(L\right)}+1}$$E84

5.5 The improved Manning-Rosen potential

To get this special case, it becomes necessary the choice $A=-2D_e\left(e^{\alpha r_e}-1\right)$, $B=0$; $C=D_e{\left(e^{\alpha r_e}-1\right)}^2,$ and $k={\alpha }^{-1}$ leading to Improved Manning-Rosen potential

$$V_{\mathit{IMR}}\left(r\right)=V\left(r\right)+D_e=D_e{\left(1-\frac{e^{\alpha r_e}-1}{e^{\mathit{\alpha r}}-1}\right)}^2+\frac{L_s}{r^2}$$E85

with, according to Eq. (49), energy spectrum given by

$${\stackrel{\sim }{E}}_{\mathit{nL}}^{\mathit{IMR}}=-{\alpha }^2{\left[\frac{2n+h_{\mathit{IMR}\left(L\right)}+1}{4}-\frac{{\alpha }^{-2}{D}_e∆E\left(e^{2\alpha r_e}-1\right)}{2n+h_{\mathit{IMR}\left(L\right)}+1}\right]}^2+D_e∆E$$E86

where

$$h_{\mathit{IMR}\left(L\right)}=\sqrt{{\left(2L+1\right)}^2+4{\alpha }^{-2}{D}_e∆E{\left(e^{\alpha r_e}-1\right)}^2},$$E87

At this point, we want to notice that in the case of $D=3$, the energy spectrum ${\stackrel{\sim }{E}}_{\mathit{nL}}^{\mathit{IMR}}$ is in agreement with Eq. (31) of Jia et al. [19] besides, it corrects Eqs. (21) and (24) of the reference [20].

On the other hand, from Eq. (52), the eigenfunctions are in this case

$${\psi }_{\mathit{nL}}^{\mathit{IMR}}\left(r\right)={\left(e^{-\mathit{\alpha r}}\right)}^{\frac{C_{\mathit{IMR}\left(L\right)}-1}{2}}{\left(1-e^{-\mathit{\alpha r}}\right)}^{\frac{h_{\mathit{IMR}\left(L\right)}+1}{2}}{{}_{2}F}_1\left(-n,b_{\mathit{IMR}\left(L\right)};c_{\mathit{IMR}\left(L\right)};e^{-\mathit{\alpha r}}\right)$$E88

with

$$b_{\mathit{IMR}\left(L\right)}=\frac{\left(h_{\mathit{IMR}\left(L\right)}+1\right)\left(h_{\mathit{IMR}\left(L\right)}+n\right)-4{\alpha }^{-2}∆E\left(e^{\alpha r_e}-1\right)-2L_s}{2n+h_{\mathit{IMR}\left(L\right)}+1}$$E89

and

$$c_{\mathit{IMR}\left(L\right)}=\frac{-2n\left(h_{\mathit{IMR}\left(L\right)}+n\right)+4{\alpha }^{-2}∆E\left(e^{\alpha r_e}-1\right)-2L_s}{2n+h_{\mathit{IMR}\left(L\right)}+1}$$E90

5.6 The Hylleraas potential

Assuming that $A=-V_0\left(1-a\right)$, $B=C=0$ and $k={\left(2\alpha \right)}^{-1}$ the potential given in Eq. (39) reduces to

$$V\left(r\right)=\frac{-V_0\left(1-a\right)e^{-2\mathit{\alpha r}}}{1-e^{-2\mathit{\alpha r}}}$$E91

for which the Hylleraas potential in D-dimensions is given by

$$V_{\mathit{Hy}}=V\left(r\right)+V_{0}a=V_0\left(\frac{a-e^{-2\mathit{\alpha r}}}{1-e^{-2\mathit{\alpha r}}}\right)+\frac{L_s}{r^2}$$E92

As before, from Eq. (51), the corresponding energy spectrum will be

$${\stackrel{\sim }{E}}_{\mathit{nL}}^{\mathit{Hy}}=\frac{-{\alpha }^2}{4}{\left[\frac{{\left(2n+h_{\mathit{Hy}\left(L\right)}+1\right)}^2+{\alpha }^{-2}{D}_e∆E{V}_0\left(a-1\right)}{2n+h_{\mathit{Hy}\left(L\right)}+1}\right]}^2+a{V}_0∆E$$E93

where

$$h_{\mathit{Hy}\left(L\right)}=2L+1,$$E94

in agreement with Hassamaadi et al. [21] when considering their parameters $b=1$ and $V_1=V_2=0$.

In relation with wavefunctions, from Eq. (52), these are

$${\psi }_{\mathit{nL}}^{\mathit{Hy}}\left(r\right)={\left(e^{-\mathit{\alpha r}}\right)}^{\frac{C_{\mathit{Hy}\left(L\right)}-1}{2}}{\left(1-e^{-\mathit{\alpha r}}\right)}^{\frac{h_{\mathit{Hy}\left(L\right)}+1}{2}}{{}_{2}F}_1\left(-n,b_{\mathit{Hy}\left(L\right)};c_{\mathit{Hy}\left(L\right)};e^{-\mathit{\alpha r}}\right)$$E95

being

$$b_{\mathit{Hy}\left(L\right)}=\frac{\left(h_{\mathit{Hy}\left(L\right)}+1\right)\left(h_{\mathit{Hy}\left(L\right)}+n\right)-2{\alpha }^{-1}∆E{V}_0\left(a-1\right)-2L_s}{2n+h_{\mathit{Hy}\left(L\right)}+1}$$E96

and

$$c_{\mathit{Hy}\left(L\right)}=\frac{-2n\left(h_{\mathit{Hy}\left(L\right)}+n\right)-2{\alpha }^{-1}∆E{V}_0\left(a-1\right)-2L_s}{2n+h_{\mathit{Hy}\left(L\right)}+1}$$E97

where, as in all the above cases, the down index indicates the name of potential, i.e. $\mathit{Hy}$ refers to Hylleraas.

5.7 The Deng Fan potential

In this case, the involved parameters are chosen as

$$A=-2b{D}_e,B=0,C=D_e{b}^2\text{and}k=1/\alpha$$E98

such that the Deng Fan potential, also called generalized Morse potential will be

$$V_{\mathit{DF}}=\frac{-2{\mathit{bD}}_e{e}^{-\mathit{\alpha r}}}{1-e^{-\mathit{\alpha r}}}+\frac{D_e{b}^2{e}^{-2\mathit{\alpha r}}}{{\left(1-e^{-\mathit{\alpha r}}\right)}^2}+\frac{L_s}{r^2}$$E99

where $D_e$ is the dissociation energy. The corresponding energy spectrum is obtained from Eq. (51) as

$${\stackrel{\sim }{E}}_{\mathit{nL}}^{\mathit{DF}}=\frac{-{\alpha }^2}{16}{\left[\frac{{\left(2n+h_{\mathit{DF}\left(L\right)}+1\right)}^2-4{\alpha }^{-2}{D}_e∆E\left(2b+b^2\right)}{2n+h_{\mathit{DF}\left(L\right)}+1}\right]}^2$$E100

which is in agreement with Oluwadare et al. [22] for the three-dimensional case, when considering the Green and Aldrich approximation [12]. Finally, from Eq. (52), the respective wave functions are

$${\psi }_{\mathit{nL}}^{\mathit{DF}}\left(r\right)={\left(e^{-\mathit{\alpha r}}\right)}^{\frac{C_{\mathit{DF}\left(L\right)}-1}{2}}{\left(1-e^{-\mathit{\alpha r}}\right)}^{\frac{h_{\mathit{DF}\left(L\right)}+1}{2}}{{}_{2}F}_1\left(-n,b_{\mathit{DF}\left(L\right)};c_{\mathit{DF}\left(L\right)};e^{-\mathit{\alpha r}}\right)$$E101

where

$$h_{\mathit{DF}\left(L\right)}=\sqrt{{\left(2L+1\right)}^2+4{\alpha }^{-2}{D}_e∆E{D}_e{b}^2},$$E102

$$b_{\mathit{DF}\left(L\right)}=\frac{\left(h_{\mathit{DF}\left(L\right)}+1\right)\left(h_{\mathit{DF}\left(L\right)}+n\right)+4{\alpha }^{-2}∆\mathit{Eb}{D}_e-2L_s}{2n+h_{\mathit{DF}\left(L\right)}+1}$$E103

and

$$c_{\mathit{DF}\left(L\right)}=\frac{-2n\left(h_{\mathit{DF}\left(L\right)}+n\right)+4{\alpha }^{-2}∆\mathit{Eb}{D}_e-2L_s}{2n+h_{\mathit{DF}\left(L\right)}+1}$$E104

At this point, it should be noted that the equivalence among the Manning-Rosen potential, the Deng-Fan and Schiöberg models for diatomic molecules have been already shown with detail in references [23, 24]. So, the solutions of the KG equation in arbitrary dimensions derived in this section can also be extended to the Schiöberg potential [25].

Finally, we want to pay attention that in a similar manner to the examples considered in this work, other exponential potentials would be achieved as particular cases from our general proposal of multi-parameter exponential-type potential [26] after a proper selection of the involved parameters.

6. Concluding remarks

Through a direct approach to transform the Schrödinger equation into a hypergeometric differential equation, we have obtained the exact solution of a class of multiparameter exponential-type potentials. Also, we have used the fact that, for equal Lorentz vector and scalar potentials, the Klein-Gordon equation can be written as a Schrödinger-type equation. With these elements, and with a proper redefinition of the involved parameters, we propose an approach to obtain the analytical solutions of the Klein-Gordon equation for exponential-type potentials, in the frame of the Green-Aldrich approximation to the centrifugal term. As a test of the usefulness of the proposed method, by an appropriate selection of parameters, the Klein-Gordon equation has been solved for specific exponential potential models such as Hulthén, Eckart, Manning-Rosen, Improved Manning-Rosen, Hylleraas and generalized Morse or Deng Fan which are derived here as particular cases from the proposal. That is, with this work, we are proposing a unified treatment for solving the Klein-Gordon equation subject to multiparameter exponential-type-potentials, leaving aside the usual methods of solution applied for each one of the aforementioned potentials, for particular parameters, given as examples. So, the displayed method offers an alternative treatment of spinless particles with new exponential-type potentials as well as the possibility to use other schemes of approximations to the centrifugal term.

Acknowledgments

This work was partially supported by the project UAMA-CBI-2232004-009. One of us (JGR) is indebted to the Instituto Politécnico Nacional-Mexico (IPN) for the financial support given through the COFAA-IPN project SIP-20231470. We are grateful to the SNI-Conacyt-México for the stipend received.