A novel exact solution of the 2+1-dimensional radial Dirac equation for the generalized Dirac oscillator with the inverse potentials

The generalized Dirac oscillator as one of the exact solvable model in quantum mechanics was introduced in 2+1-dimensional world in this paper. What is more, the general expressions of the exact solutions for these models with the inverse cubic, quartic, quintic and sixtic power potentials in radial Dirac equation were further given by means of the Bethe ansatz method. And finally, the corresponding exact solutions in this paper were further discussed.


Introduction
As we all know, Dirac oscillator as a concept which was first proposed by M. Moshinsky [1] and collaborators by means of the combination between momentum and coordinate p p im r   , where r is the coordinates, m represents the mass of particle, and  is regarded as the frequency of particle [2][3]. Although the Dirac oscillator, an effective relativistic model, has been succeed in describing the interaction between anomalous magnetic moment and linear coordinates in many fields, it is powerless to explain the interaction between anomalous magnetic moment and nonlinear coordinates. Therefore, in order to describe the complex interactions, Dutta and his colleagues [4] proposed a new concept which is called the generalized Dirac oscillator. So it could generalized the linear effect between coordinates and momentum to nonlinear effect by using this system, that is to say, the Dirac oscillator could be regarded as a special case in the generalized Dirac oscillator.
As one of the few relativistic quantum systems that can be solved accurately, Dirac oscillator has attracted a lot of attentions [1][2][3][4][5]. This system can not only solve the quark confining potential [5][6][7] in quantum chromodynamics, but also can be used to solve some complex interactions [8][9]. Considering the practicability of solving problems, Dirac oscillator has been extensively discussed by many researchers in various aspects: like conformal invariance properties [8], covariance properties [9], shift operators [10], symmetry Lie algebra [11], hidden supersymmetry [12][13][14], completeness of wave functions and so on [15][16]. Moreover, many phenomena in Quantum Optics [17][18][19][20][21] could also be explained by this model. The investigation of the relationship between Dirac oscillator and relativistic Jaynes-Cummings model [4,[20][21] bridges the two unrelated fields of Relativistic Quantum Mechanics and Quantum Optics [22][23]. In addition, this model can also be used to explain some new phenomena in condensed matter physics [16][17][24][25], such as the Quantum Hall Effect and Fractional Statistics. So it is interesting to extend Dirac oscillator to generalized Dirac oscillator for solving the interaction between momentum and coordinate in nonlinear electric field.
To solve 1+1-dimensional Dirac equation with complex interaction potential, Dutta [26][27][28][29] provided that appropriate selection the interactions. Next, the structure of this paper is as follows. The 2+1-dimensional generalized Dirac oscillator is introduced in Section 2. In Section 3, exact solutions of the radial Dirac equation with generalized Dirac oscillator are given by using the functional of Bethe ansatz method [26][27][28][29] when selecting appropriate interactions. Section 4 is devoted to the conclusion. And finally, Bethe ansatz method will be briefly reviewed in appendix A.

2+1-dimensional generalized Dirac oscillator
For particle of mass where the matrices and the eigenfunction satisfy the following conditions According to the equation (4), the equation (5) can be written as follow So, the second order differential equation could be given by substituting the lower formula in equation (6) . Next, the exact solution of this system will be further discussed in following section by using the Bethe ansatz method.

Explicit implementation of generalized Dirac oscillator
Over the past decades, the singular potential has attracted lots of attentions because it can be used to describe many physical problems. For example, the singular potential is not only widely applied in the research of the   , pp and   , p  procedures in high-energy physics [29,[34][35], but also repulsive singular potentials could reproduce the interactions of   scattering and nucleons with K-mesons [29,36]. Moreover, high energy scattering question caused by strong singular potential, as an example of non-relativistic quantum mechanics, has also been extensively discussed by many authors [29,[37][38]. Besides, the singular potentials were also widely used to describe the interaction of two atoms in molecular physics, inter-atomic or intermolecular force and chemical physics [29]. Considering the wide applications of the singular potential, the exact solutions of 2+1-dimensional radial Dirac equation with generalized Dirac oscillator under the inverse cubic, quartic, quintic and sixtic power potentials were discussed in this paper. And it is also shown that the conclusion of the generalized Dirac oscillator with higher order inverse potential can degenerate to the conclusion of lower order inverse power potential interaction when the parameters are properly selected. Next, the analytical expressions of the corresponding exact solutions will be given in section 3, the ground and first excited state was discussed in more detail.

Quasi exact solution of the inverse cubic
power potential Now, let's firstly discuss the inverse cubic power potential This potential has been investigated in Schrodinger equation by the other authors to obtain the accurate analytical expression [39][40]. Now, in order to solve this problem in the 2+1-dimensional radial Dirac equation, the corresponding radial Dirac equation could be given by substituting equation (8) into equation (7)     The exact solution of the 2+1-dimensional radial Dirac equation is further given by extracting the asymptotic behavior of the wave function   , r  through a simple replacement. Next, by making a brief examination for differential equation, we implement transformation here the parameters ,, AB  and D are constant and m is magnetic quantum number. Substituting equation (10) into differential equation (9), it's easy to find that parameters satisfy underlying relations Therefore, new differential equation could be written as follow here parameters satisfy 1 the equation (11) could also be solved accurately by using the Bethe ansatz method [26][27][28][29] if the potential parameters satisfy certain constraints. Now, in order to realize the square integrable of wave functions, we assume that The above wave functions are square integrable, so the corresponding normalization constants could also be given through the standard integral [ Obviously, if the potential parameters satisfy certain constraints, the exact solution of the system could be given by using the Bethe ansatz method [26][27][28][29]. And the condition of the wave function has an acceptable asymptotic behavior is that the parameter B must be negative when r .

Quasi exact solution of the inverse quartic power potential
The interaction of the Dirac equation with inverse quartic power potential is studied in this part From the phenomenological point of view, singular potential as a very useful form of anharmonicity is used in many aspects of physics [42][43]. The inverse quartic potential has also been investigated in many different questions by lots of authors [44][45][46]. Now, the purpose of our study is to obtain the analytical properties of the scattering amplitude about singular potential. So, the corresponding 2+1-dimensional radial Dirac equation with inverse quartic power potential is given by equation (7)   In order to deal with the question, the appropriate asymptotic behavior of the wave function    (21) where the parameter is set 2 The exact solvable form of the radial Dirac equation with inverse quintic power potential could be further given by substituting equation (27) into equation (7 The wave functions are also squarely integrable:   2 0 n r dr     . Next, the ground and first excited state will be discussed in detail as two special