Approximate Complete Solutions of DKP Equation under a Vector Exponential Interaction via a Pekeris-Type Approximation

The DKP equation for an exponential potential is exactly solved via an appropriate approximation using the methodology of supersymmetric quantum mechanics. We see that the solutions are already known without any cumbersome algebra we face in any numerical or analytical approach. Closed forms of eigenfunctions and eigenvalues are reported.


Introduction
Working on the basis of Dirac and Klein-Gordon equations, spin-0 and spin-1/2 particles have been extensively discussed via many analytical and numerical techniques. For the spin-1 particles, however, there are only few investigations. The main reason for this lack of literature is probably the mathematical complexity of Proca equation describing the spin-1 particles. The Duffin-Kemmer-Petiau DKP 1-4 equation, however, provides us with a theoretical basis for describing both spin-0 and spin-1 particles on a relatively easier background. For many years, the DKP equation was thought to be exactly equivalent to Klein-Gordon KG and Proca equations and consequently was in the shadow of them. Now, we know that the equations are not exactly the same and the equivalence does not hold generally 5-14 . In addition, the DKP equation is richer in the investigation of interactions and is even closer to some experimental data in comparison with KG or Proca equations 15-20 . Moreover, besides cosmology and gravity, this equation has been tested in many branches of physics including particle and nuclear physics 21-25 . As usual, the most appealing case studies are Coulomb and quadratic terms 26-28 , and other ones including the woods-Saxon and Hulthen are investigated by different 2 ISRN High Energy Physics approaches as well [29][30][31][32] . Within the present work, we first review the DKP equation. Next, an introductory section is included on supersymmetry SUSY quantum mechanics. In the last part, we obtain the approximate analytical solutions of the problem.

DKP Equation
For the sake of briefness, our starting square is with the upper and lower components, respectively, being

2.3
β 0 is the usual 5×5 matrix, and U s , U o v denote the scalar and vector interactions. The equation, in 3 0 -dimensions, is therefore written as It should be noted that ψ is a simultaneous eigenfunction of J 2 and J 3 , that is

and the general solution is
where the 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 denote the radial wave functions. It is shown that the above equations result in the coupled differential equations 26, 33 : where α J J 1 / 2J 1 and ζ J J/ 2J 1 . In the case of U s 0, we recover the well-known formula Within the next section, we give a brief introduction to the SUSY method.

SUSY and Shape Invariance
The basic idea of the SUSY approach is based on finding the solutions of the Riccati equation with V being the potential of Schrödinger equation. If the condition is satisfied, we call the partner Hamiltonians shape invariant. In the latter relation, a 1 is a new set of parameters uniquely determined from the old set a 0 via the mapping F : a 0 → a 1 F a 0 and the residual term R a 1 does not include x. Actually, the shape invariance implies that the partner potential, apart from some constant terms, is interpreted as a new partner potential V − a 1 , x associated with a new SUSY potential Φ a 1 , x 34 . In such a case, the problem is simplified to a high degree and everything of interest is calculated via the elegant idea of 34-36 : dzΦ a n , z .

2.14
That is, the energy eigenvalues as well as the corresponding eigenfunctions are obtained without cumbersome algebra, the Hence, shape invariance gives the complete and exact information about the spectrum of the bound states of the following Hamiltonians: In other words, everything is obtained without any cumbersome algebra provided that the superpotential is simply found and the shape invariance exists.

A Famous SUSY Example
Within this section, we review an SUSY example. The results can be found in 34-36 . Choosing the superpotential as

2.20
Obviously, the partner potentials are shape invariant via a mapping of the form For the energy relation, as

Approximate Analytical Solution of the Radial Part
We now see that the problem appears as the latter SUSY problem. Choosing the potential as U v V e −a r−r 0 , we get We wish to emphasize that, on the one hand, exponential-type potentials find application in a wide class of physical sciences including cosmology 37-40 , nuclear and particle physics 41-45 , solid state 46-48 , atomic and molecular physics 49-58 , and chemical physics 59, 60 . On the other hand, the DKP equation itself, as an outstanding relativistic wave equation which well explains both microscale phenomena in particle physics and large-scale ones in cosmology, motivated us to do the present calculations. Naturally, depending on the system under consideration, the phenomenological fits are quite different. To be able to analytically solve the problem, let us use the well-known approximation 61 3.3c ISRN High Energy Physics 7 We find which is a Schrödinger-like equation corresponding to the Morse potential. Before proceeding further, it should be noted that the choice of the system under consideration definitely puts limitations on the approximation 3.2 via 3.3a , 3.3b and 3.3c to yield acceptable results.
Here, however, as we intend to give a general background for related studies, we avoid focusing on a special system since the numerical data are quite different from one system, for example, a diatomic molecule, to another such as cosmological model. Also, there exist many other papers which use Pekeris-type approximations for Schrödinger, Dirac, and KG equations for a lengthy list of potentials including the Hulthen, Eckart, Rosen-Morse, and Pöschl-Teller see 62-69 and references therein .
In addition, the interested reader might find attractive discussion on the SUSY structure of DKP equation in 70 . Moreover, we wish to address the interesting papers 71-

Conclusion
We obtained approximate analytical solutions of the DKP equation for an exponential term which in many cases provides more exact solutions than linear or quadratic terms. The results are applicable to many branches of physics from cosmology to particle physics.