Solutions of the Klein Gordon equation with generalized hyperbolic potential in D-dimensions

We solve the D- dimensional Klein–Gordon equation with a newly proposed generalized hyperbolic potential model, under the condition of equal scalar and vector potentials. The relativistic bound state energy equation has been obtained via the functional analysis method. We obtained the relativistic and non-relativistic ro-vibrational energy spectra for different diatomic molecules. The numerical results for these diatomic molecules tend to portray inter-dimensional degeneracy symmetry. Variations of the energy eigenvalues obtained with the potential parameters have been demonstrated graphically. Our studies will find relevant applications in the areas of chemical physics and high-energy physics.


Introduction
Researchers over the years have continually sought for solutions of wave equations with potential energies both in the non-relativistic and relativistic quantum mechanical systems [1,2]. These solutions will provide all the necessary information needed to explain the behavior of any physical system. In addition, the solutions of these wave equations are highly applicable in chemical physics and high-energy physics at higher spatial dimensions [3]. Klein-Gordon (KG) equation is a basic relativistic wave equation that is well known to describe the motion of spin zero particles [4]. Different investigations have been carried out to obtain the exact and approximate solutions of the KG equation with different potentials, via various methods including the asymptotic iteration method (AIM) [5], Nikiforov-Uvarov (NU) method [6], supersymmetric quantum mechanics (SUSYQM) [7], algebraic approach [8], exact and proper quantization rules [9], modified factorization method [10,11] and others [12][13][14][15][16].Many authors have studied the solutions of the D-dimensional Klein-Gordon equation with diatomic molecular potential energy models [17][18][19][20][21][22][23][24][25]. Analytical solutions of the KG equation and Dirac equation have been obtained for the conventional form of the Rosen-Morse (RM) potential energy model [26,27]. Chen and his collaborators [28] studied the relationship between the D-dimensional relativistic rovibrational energies with applications to the Lithium diatomic molecule. In addition, RM type scalar and vector potential energy model was employed to obtain the s-wave bound state energy spectra [29]. Villalba et al [30] considered the bound state solution of a one-dimensional Cusp potential model, confined in the KG equation. The bound state solution of the KG equation with mixed vector and scalar PT potential energy with a nonzero angular momentum parameter was investigated by Xu et al [31]. Badalov et al [32] used NU to study any l-state of the KG equation, with the help of a Pekeris-like approximation scheme. In similar development, Ikot et al [33] solved the KG equation with the Hylleraas potential model and obtained its exact solution. Also, Hassanabadi and his collaborators [20] studied a combined Eckart potential and modified Hylleraas potential energy in higher dimensional KG equations using supersymmetric quantum mechanics method. Jia et al [22] investigated the bound state solution of the KG equation with an improved version of the Manning-Rosen potential model. Ortakaya [34] solved the D-dimensional KG equation and obtained the bound state energy spectrum for three different diatomic molecules using pseudoharmonic oscillator potential model. Chen et al [28] employed the improved MR potential energy in D-spatial dimensions to obtain the relativistic bound state energy equation. Also, Ikot et al [35] analyzed the improved MR potential energy for arbitrary angular momentum parameter using an approximate method in D-dimensions. Xie et al [36] studied the bound state solutions of the KG equation with the Morse potential energy in D-spatial dimensions. Ikot and his co-authors [37] employed NU method to investigate the D-dimensional KG equation with an exponential type molecule potential model. Hyperbolic potential models have been used as the empirical mathematical models in describing various interatomic interactions for diatomic and polyatomic molecules [38]. Deformed hyperbolic functions have also been studied and its non-relativistic energy spectra obtained via different methods [39][40][41][42][43][44].Most recently, Durmus [45] studied the Dirac equation with equal scalar and vector hyperbolic potential function using the AIM, with the help of Greene and Aldrich approximation scheme. The author also investigated the relativistic vibrational energy spectra for various electronic states of some alkali metal diatomic molecules.Motivated by the work of Durmus [45], we propose a generalized hyperbolic potential (GHP) of the form 4 are potential parameters, and a is the range of the potential. Using the functional analysis method, we investigate the approximate bound state solution of the KG equation with GHP in higher spatial dimensions. We also explore the properties of the D-dimensional relativistic and non-relativistic ro-vibrational energy spectra for the GHP analytically and numerically for some selected diatomic molecules.

Bound state solutions
The Klein-Gordon equation with a scalar potential ( ) S r and a vector potential ( ) where D represents the spatial dimensionality and   D 2, D 2 represents the Laplace operator in D-dimensions,  is the reduced Planck constant, c and E are the speed of light and relativistic energy of the system, respectively. Also, the wave function can be given as Y W = W -- where we write the radial part of the D-dimensional Klein-Gordon equation (2) as E vJ represents the relativistic ro-vibrational energy eigenvalues in D-dimensions, v J and represents the vibrational and rotational quantum numbers, respectively. For equal scalar and vector potentials, Rescaling the scalar potential ( ) S r and vector potential ( ) V r under the non-relativistic limit, we adopt the Alhaidari et al [47] scheme to write equation (4) as With the equal scalar and vector potential being taken as the generalized hyperbolic potential, we obtain the following second-order Schrodinger-like equation as, Due to the presence of the centrifugal term in equation (7), we employ the Greene-Aldrich approximation scheme [48] a a a = -= As noted in [45], the above approximation is seen to be valid only for short range potential with small potential range, a. This approximation tends to break down for large a. Substituting equation (8) and introducing coordinate transformation of the form a = ( ) s r tanh , 2 we get e -+ - Also, we propose the wave function as We find that equation (9) The solution of equation (16) can be expressed in terms of the hypergeometric function given below where To obtain the energy relation, we equate either equations (19) or (20) to a negative integer (sayv). Hence, we choose Substituting equations (10)- (12), (14), (15) and (17) into (22), we obtain the D-dimensional relativistic rovibrational energy spectra for the GHP in the form To obtain the nonrelativistic ro-vibrational energy spectra for the GHP, we employ the following mapping: and .  The normalization of the wave function can be determined as shown in appendix appendix.

Results and discussion
We consider different diatomic molecules (HCl NiC CO I , , , 2 ) with spectroscopic parameters as shown in table 1. These parameters were adopted from [49] and applied to equation (24) to compute the numerical values of the non-relativistic ro-vibrational energies for arbitrary quantum numbers in different dimensions, as shown in tables 2-5. We observe from the tables presented that the non-relativistic ro-vibrational energies for the selected diatomic molecules decrease as the quantum numbers (v J , ) increase. Also, for any quantum state, there is a decrease in ro-vibrational energies as the dimension increases. This trend is consistent with the relation of energy eigenvalues and quantum numbers, as observed in [49] for the selected diatomic molecules. In addition, we observe that there exist an inter-dimensional degeneracy symmetry for the selected diatomic molecules . This implies that the nonrelativisticro-vibrational energy spectra for the GHP is invariant under a transformation of an increase in the D-dimension by two (  + D D 2) and a decrease in the rotational quantum number by one (  -J J 1).
The result in equation (27) is very consistent with [45,50]. This results' accuracy have been tested by calculating the ro-vibrational energy spectra of the equation (27) numerically for different quantum states and various potential range,a. We have compared our result with other results obtained using different methods such as AIM [45] and algebraic method [50], as shown in table 6. We also set = = V V 0 1 3 to have hyperbolic Rosen-Morse potential from the GHP in the form The non-relativistic ro-vibrational energy spectra of the hyperbolic Rosen-Morse potential is obtained to be   a m m a = -+  We also plot the graphs of the non-relativistic ro-vibrational energies with respect to the potential range, dimensions, rotational and vibrational quantum numbers, potential parameters, as shown in figures 1-8, respectively. From figures 1-4 respectively, it is seen that there is a monotonic decrease in the non-relativistic energies as a D J v , , , and increases for the selected diatomic molecules. Figures 5 and 6 show the increase in E vJ as the potential parameters V V and 1 2 increases, respectively. In figures 7 and 8, the non-relativistic rovibrational energies increases to a peak value and later decreased as the potential parameters V V and 3 4 increases, respectively. In addition, we considered the variation of E vJ with spatial dimension D for various quantum states of HCl molecule as shown in figure 9. As the spatial dimension increases, the non-relativistic ro-vibrational energy E vJ decreases slowly and later decreases in a monotonic manner. Figure 10 shows a sharp decrease in E vJ as the vibrational quantum number increases for different spatial dimensions of HCl molecule.

Conclusion
In our study, we solve the D-dimensional Klein-Gordon (KE) equation with our newly proposed generalized hyperbolic potential (GHP) model using the functional analysis method. By employing the Greene-Aldrich-like approximation scheme, we obtain an expression for the D-dimensional relativistic ro-vibrational energy spectra  for the GHP. Also, this expression was reduced to the non-relativistic case by employing the necessary mapping scheme. Numerical results for the D-dimensional non-relativistic ro-vibrational energy spectra were obtained for different diatomic molecules (HCl NiC CO I , , , 2 ), for arbitrary quantum numbers. Special cases were obtained where our results agree with the results obtained in the literature. Our results for different diatomic molecules show inter-dimensional degeneracy symmetry as the dimensions increase and the rotational quantum number decreases. Different plots of non-relativistic ro-vibrational energy spectra versus the GHP parameters were also analyzed and discussed. These plots show a monotonic decrease in the energy eigenvalues as the potential parameters increase for the diatomic molecules considered. A specific consideration was given to HCl molecule, as the variation of its non-relativistic ro-vibrational energy eigenvalues with both D-spatial dimension and vibrational quantum numbers, respectively, were discussed.     The standard integral is given as [32] ò -+ = G + + G + + G + + +