Quasi-Analytical Solutions of DKP Equation under the Deng-Fan Interaction

Quasianalytical solutions of Duffin-Kemmer-Petiau equation under scalar and vector Deng-Fan potentials are reported via a novel ansatz.


Introduction
A yet open challenge in the annals of wave mechanics is the equivalence or nonequivalence of Duffin-Kemmer-Petiau DKP and Klein-Gordon KG equations.While the latter solely describes relativistic spin-zero bosons, the former can investigate spin-one particles as well.This equation was introduced 1930s in search for a linear relativistic wave equation similar to Dirac equation for relativistic bosons 1-4 .In its present format, the DKP equation is normally represented in two five and ten-dimensional versions that respectively work for spin-zero and spin-one bosons.At the present, to our best knowledge, the equivalence of KG and spin-zero DKP equations is doubted 5-14 .This is also true for the other counterpart of DKP equation, that is the Proca equation which is a relativistic framework to study spin-one bosons.There are papers that investigate related problems via these equations and report motivating data 15-20 .Here, our focus is on the spin-zero version of the equation, which has many applications 21-25 .
In solving wave equations of mechanics, various analytical techniques including supersymmetry quantum mechanics SUSYQM , Nikiforov-Uvarov NU , quantization rule, Lie algebras, WKB approximation, point canonical transformation PCT , and series expansion have been applied 26-40 .Nevertheless, we face situations in which these methodologies do not work very well.A very successful approach in such cases is proposing

The DKP Equation
The DKP Hamiltonian for scalar U s and vector U o v interactions is where and the upper and lower components, respectively, are

2.5
where A A 1 , A 2 , A 3 .In 2.4a and 2.4b , ψ is a simultaneous eigenfunction of J 2 and J 3 , that is, and the general solution is considered as where spherical harmonics Y JM Ω are of order J, Y M JL1 Ω are the normalized vector spherical harmonics, and f nJ , g nJ , and h nJL represent the radial wavefunctions.The above equations yield the following coupled differential equations 30-38 : Advances in High Energy Physics which give 39-41 where When U s 0, we recover the well-known formula 30-32 We consider the Deng-Fan vector and scalar potentials 42-46

2.11
By a change of variable of the form 2.9 is written as

2.15
Here, we use the following approximations for the centrifugal term 47 : Equation 2.16a is a quite logical alternative for α < 0.1 see Figure 1 , and 2.16a and 2.16b brings 2.15 into the form e αr e αr − 1 3

2.17
By introducing z e αr and making the transformation u n,J z φ n,J z / √ z , we obtain which after decomposition of fractions gives Advances in High Energy Physics  with

2.20
Let us now consider an ansatz of the form

2.22
Substitution of the latter in 2.20 yields Advances in High Energy Physics

2.23
For the sake of simplicity, here we consider only the nodless solutions.Substitution of the proposed ansatz solution and equating the corresponding powers on both sides give

2.24
For the fixed values of V 1 , u 1 , α, and m, in particular, the system of ten equations 2.24 determines the sets of variables E 0,J , V 0 , V 2 , u 0 , u 2 , γ, β, ξ, η, and δ.Therefore, the spectrum and eigenfunctions of the system are easily obtained for a particular system.For the higher states, the mathematical process is more cumbersome and complicated but can be followed by the same token we did here, that is, by choosing f 1 z z−α 1 1 for the first node, f 2 z z − α2 1 z − α 2

Conclusion
Motivation behind our study was the high number of spin-zero relativistic systems that we frequently face as well as the attractive structure of the Deng-Fan potential.The corresponding ordinary differential equation was too complicated to be solved by common analytical techniques.We, therefore, performed some novel transformations and applied an acceptable approximation to the centrifugal term.We next proposed an interesting physical ansatz solution by which we were able to find a quasi-analytical solution.Our results are particularly useful in particle and nuclear physics and can be directly used after prerequisite fits performed.

Figure 1 :
Figure 1: 1/r 2 and its approximation for different value of α.