Approximate scattering state solutions of DKPE and SSE with Hellmann Potential

We study the approximate scattering state solutions of the Duffin-Kemmer-Petiau equation (DKPE) and the spinless Salpeter equation (SSE) with the Hellmann potential. The eigensolutions, scattering phase shifts, partial-waves transitions and the total cross section for all the partial waves are obtained and discussed. The dependence of partial-waves transitions on total angular momentum number, angular momentum number, mass combination and potential parameters were presented in the figures.


Introduction
The quantum mechanical study of Hellmann potential is a time-honoured and prominent problem.

Just of recent, Yazarloo et al. extended the study to scattering states of Dirac equation with
Hellmann potential under the spin and pseudospin symmetries [4]. The Dirac phase shift and normalized for the spin and pseudospin symmetries wave function were reported. Also, in one of our previous papers, we studied the scattering state solution of Klein-Gordon equation with Hellmann potential [5].
The motivation behind this work is to investigate the approximate scattering state solutions of

Duffin Kemmer Petiau equation (DKPE) and spinless Salpeter equation (SSE) with Hellmann
Potential. The SSE explains in detail the dynamics of semi-relativistic of particle [6-10 and the references therein] and two-body effects whereas the DKPE explains explicitly the dynamics of relativistic spin-0 and spin-1 particles [11][12][13][14][15][16][17][18]. The Hellmann potential in this study may be written as [1][2][3][4][5] where and are the strengths of the Coulomb potential and Yukawa potential, respectively and is the potential screening parameter which regulates the shape of the potential.
Section 2 presents scattering state solutions of DKPE with Hellmann potential. The scattering state solution of SSE with Hellmann potential is presented in Section 3. In Section 4, we discuss the results and the conclusion are given in Section 5.

Scattering states of the Duffin-Kemmer-Petiau equation (DKPE) with Hellman potential
The DKP equation with energy , , total angular momentum centrifugal term and the mass m of the particle is given as [11][12][13][14]: , (2) The effect of total angular momentum centrifugal term in Eq. (2) can be subdued using approximation scheme of the type [7, 10, 14] The above approximation has been reported to be valid for ≪ 1 [14]. By substituting Eqs. (1) and (3) into Eq. (2) and transform using mapping function = 1 − − , leads to where we have employed the following parameters for simplicity and = √( , 2 − 2 ) + 2 2 − 2 , − ( + 1) 2 is the wave propagation constant. Choosing the trial wave function of the type: and substituting it into Eq. (4), we obtain the hypergeometric type equation [19] (1 − ) , Therefore, the DKP radial wave functions for any arbitrary − states may written as: where , is the normalization factor. The phase shifts and normalization factor , can be obtained by applying the analyticcontinuation formula [19].
Accordingly, with the appropriate boundary condition imposed by Ref. [20], Eq. (22) yields and on comparison of Eq. (22), the DKP phase shift and the corresponding normalization factor can be found respectively as: and The DKP total cross-section for the sum of partial-wave cross-sections is defined as [10]: (30)

Scattering states solutions of the spinless Salpeter equation (SSE) with Hellmann potential
The spinless Salpeter equation for two different particles interacting in a spherically symmetric potential in the center of mass system is given by [21][22][23][24]:  The total scattering cross-section for the sum of partial-wave cross-sections is given as where = 4 2 (47) which defines the partial-wave transitions for the SSE with Hellmann potential in this present study.

Discussion
We have used the units ħ = = 1 in partial-wave transition illustrations. For equal mass cases, we used ( / ) 3 = 1/4 and = 1 /2 while ( / ) 3 = 1 and = 1 100 were used for unequal masses cases. In all the cases, we consider 2 = , = 1 and 1 = 1 for the equal masses case only. For the screening parameters = 0.1, = 0.2 and = 0.3 the DKP partial-waves transitions increases exponentially. See Figure 1. The two-body effect here appears as a shift of the phases of the partial waves. For lower partial-waves, says < 5, the partial-waves transition decay exponentially whereas for higher partial waves, says > 5, the partial-waves transition rise exponentially. See

Conclusion
We have investigated the approximate scattering state solutions of DKPE and SSE with Hellman potential via analytical method. The approximate DKP and semi-relativistic scattering phase shifts, partial-wave transitions, eigenvalues and normalized eigenfunctions have been obtained. The DKP and semi-relativistic partial wave transition calculations for the Hellmann potential were shown in the Figures 1 and Figures 2-4 respectively. It is clearly shown both the total angular momentum number, angular momentum number and potential parameters contribute significantly to the partial wave transition and that the two-body effects modify the phases of the partial-waves and is usually noticeable for lower partial-waves.