Generalized Dirac oscillator in cosmic string space-time

In this work, the generalized Dirac oscillator in cosmic string space-time is studied by replacing the momentum pu with its alternative p_u+mwbf_u(x_u)). In particular, the quantum dynamics is considered for the function f_u(x_u) to be taken as cornell potential, exponential-type potentialand singular potential. For cornell potential and exponential-type potential, the corresponding radial equations can be mapped into the confluent hypergeometric equation and hypergeometric equation separately. The corresponding eigenfunctions can be represented as confluent hypergeometric function and hypergeometric function. The equations satisfed by the exact energy spectrum have been found. For singular potential, the wave function and energy eigenvalue are given exactly by power series method.

oscillator has many applications and has been studied extensively in different field ssuch as high energy physics [12][13][14][15],condensed matter physics [16][17][18], quantum Optics [19][20][21][22][23][24][25] and mathematical physics [26][27][28][29][30][31][32][33] etc. On the other hand, the analysis of gravitational interactions with a quantum mechanical system has recently attracted a great deal attention and has been an active field of research [5][6][34][35][36][37][38][39][40][41][42][43].The study of quantum mechanical problems in curved space-time can be considered as a new kind of interaction between quantum matter and gravitation in the microparticle world. In recent years, the Dirac oscillator embedded in a cosmic string background has inspired a great deal of research such as the dynamics of Diracoscillator in the space -time of cosmic string [44][45][46][47], Aharonov-Casher effect on the Dirac oscillator [5,48], noninertial effects onthe Dirac oscillator in the cosmic string space-time [49][50][51]etc. It is worth mentioning that based on the coupling corresponding to the Dirac oscillator a new coupling into Dirac equation first has been proposed by Bakke et al [52]. and used in different fields [53][54][55][56][57]. This model is called the generalized Dirac oscillator which, in special case is reduced to ordinary Dirac oscillator. Inspired by the above work, the main aim of this paper is to analyze the generalized Dirac oscillator model with the interaction functions f μ (x μ ) taken as cornell potential, singular potential and exponential-type potential in cosmic string space-time and to find the corresponding energy spectrum and wave functions. This work is organized as follows. In sect.2, the new coupling is introduced in such a way that the Dirac equation remains linear in momenta, but not in spatial coordinates in a curved space-time. In sect.3, Sect.4 and Sect.5, we concentrate our efforts inanalytically solving the quantum systems with different function f μ (x μ ) and find the corresponding energy spectrum and spinors respectively. In sect.6, we make a short conclusion.

2、Generalized Dirac oscillator with a topological defect
In cosmic string space-time, the general form of the cosmic string metric in cylindrical coordinates read [41,42,44,58,59] , (2) where the   matrices are the generalized Dirac matrices defining the covariant , m is mass of the particle, and   is the spinor affine connection. We choose the basis tetrad  then in this representation the matrices γ μ [44] can be found to be γ 0 = γ t ,γ 1 = γ ρ = γ 1 cos φ + γ 2 sin φ , It is well known that in both Minkowski spacetime and curved spacetime, usual Dirac oscillator can be obtained by the carrying out non-minimal substitution p μ ⟶ p μ + mωβx μ in Dirac equation where m and ω are the mass and oscillator frequency. In the following, we will construct the generalized oscillatorin curved spacetime. To do this end, we can replace momenta p μ in the Dirac equation of curved spacetime by where f μ (x μ ) areundetermined functions of x μ . It is to say that we introduct a new coupling in such a way that the Dirac equation remains linear in momenta but not in coordinates. In particular, in this work, we only consider the radial component the non-minimal substitution f μ (x μ ) = (0, f ρ (ρ), 0,0).
With help of the equation (12) and simple algebraic calculus the equation ( It is easy to prove the following relation [44] iσ 1 σ 2 λ+ikρ(σ 1 σ 3 cos φ + σ 2 σ 3 sin φ) = −2s ⃗. L ⃗⃗ , where s ⃗ = For the componentχ 2 , from equation (10) an analog equation can be also obtained In particular, the equation (14) will reduced to the result obtained in Ref. [44] when 3、The solution with ( ) to be cornell potential The cornell potential that consists of Coulomb potential and linear potential, has gotten a great deal of attention in particle physics and was used with considerable success in models describing systems of bound heavy quarks [60][61][62]. In cornell potential, the short-distance Coulombic interaction arises from the one-gluon exchange between the quark and its antiquark, and the long-distance interaction is included to take into account confining phenomena.
Now we let the function f(ρ) to be cornell potential where a and b are two constants. Substituting (17) into (*a) and (*b) leads to following equation where τ 1 2 = λ 2 − μb + m 2 ω 2 b 2 − ωmb, We make a change in variables ξ = mωaρ 2 and then the equation (18) can be rewritten as ξ d 2 χ Taking account of the boundary conditions satisfied by the wave function χ, i.e., χ ∝ ξ τ 1 2 ⁄ for ξ → 0 andχ ∝ e − ξ 2 for ξ → ∞, physical solutions χ can be expressed as [44,60,63] If we insert this wave function χ into Eq. (20), we have the second-order homogeneous linear differential equation in the following form: It is well known that the equation (22) is the confluent hypergeometric equation and it is immediate to obtain the corresponding eigenvalues and eigenfunctions with δ 1 = m 2 + k 2 − 2abm 2 ω 2 + mωa, In particular, if we assume that α = 1, from equation (24), the energy levels of generalized Dirac oscillator with f(ρ) to be cornell potential, in the absence of a topological defectcan obtained. In addition if we let a = 1,b = 0 in equation (24) the energy levels given here will bereduced to that one obtained in reference [44].
4．The solution with ( ) to be singular potential The investigation of singular potentials in quantum mechanics is almost as old asquantum mechanics itself and covers a wide range of physical and mathematical interest because the real world interactions were likely to be highly singular [64].
Thesingular potentials of v(r) ∝ 1 r n type, with n ≥ 2 is of great current physical interest and is relevant to many problems such as the three-body problem in nuclear physics [65][66], point-dipole interactions in molecular physics [67], the tensor force between nucleons [68], and the interaction between a chargesand an induced dipole [69] respectively. Recently, in cosmic string background, singular inverse square potential with a minimal length had been studied [70].
From the recursion relation(36) we can determinethecoefficientsa n ( ≠ 0,1) of the power series in terms of a 0 and a 1 . In addition the above recursion relation implies that equation (35) yields one solution as a power series in even powers of ρ and another in odd powers of ρ.
In addition, the equation (27) can be also mapped to the double-confluent Heun equation by appropriate function transformation [75]. So when f(ρ) is taken as singular inverse-square-type potential, the solutions of the equation (27) can been also given by the solution of the double-confluent Heun equation [75,76].

5、The solution with ( ) to be exponential-type potential
The exponential-type potentials are very important in the study of various physical systems, particularly for modeling diatomic molecules. The typical exponential-type potentials include Eckart potentials [77], the Morse potential [78,79], the Wood-Saxon potential [80] and Hulthé n potential [81,82] etc. The research work on the Dirac equation with the above potential is mainly concentrated on Minkowski time and space. However, it has been noticed recently that it is also interesting to stusy this kind quanstum systems in a cosmic string background [83]. In this section we will take the f(ρ) as exponential-type functionand solve the corresponding Dirac equation in cosmic string space-time.
As is known to all，the Dirac equation and Schrödinger equation have been studied by resorting different methods. A usual way is transforming the eigenvalue equation of quantum system considered into a solvable equation via suitable variable substitutions and function transformations [84][85][86]. In order to obtain solution forf(ρ) being exponential-type potential, we firstly consider the following linear second-order differential equation where ℒ i , (i = 1,2,3) are constants. It is known that singular points of a differential equation determine the form of solutions. In this equation, there are two singular points, i.e.,x = 0 and x = 1. In order to remove these singularities and get physically acceptable solutions we use the following ansatz: where Λ and Ω are two real parameters. Further we make this two parameters to satisfy following relationships and by substitution the equation (38) to the equation (37), the differential equation forχ(x) can be written as with Δ = ±√−ℒ 3 .In other words, the equation (37) and the corresponding eigenfunctions is given in terms of the Gauss hypergeometric functions below R(x) = AF(τ 1 , τ 2 ; 1 + 2Ω; x), where A is normalization constant. Next , we will use the results given here to obtain the solutions of Dirac equation exponential-type interaction in cosmic string space-time. As a direct application of the above method, let us take the function f(ρ) to be as Yukawa potential, Hulthé n-Type potential and generalized Morse potential respectively.

Case 1. ( ) being Yukawa potential
In Yukawa meson theory, the Yukawa potential firstly was introduced to describe the interactions between nucleons [88]. Afterwards, it has been applied to many different areas of physics such as high-energy physics [89,90], molecular physics [91] and plasma physics [92]. In recent years, the considerable efforts have also been made to study the bound state solutions by using different methods.
Now let us choose f(ρ) to be Yukawa potential However, the radial equation (44) cannot accept exact solution due to the presence of the centrifugal term [85]. In order to find analytical solution, we have to usesome approximation approaches for the centrifugal term potential. Following reference [86], the approximation for the centrifugal term reads: It's worth mentioning that the above approximations are valid for βρ ≪ 1 [86]. So If we make the controlparameter β small enough, then we can guarantee that the above approximations in equation (45)holds for larger values ρ . In other words, this approximation(45) is valid in our work.
It is easy to see that the differential equation (60) is also similar to the equation (37).
The above results show that the radial equation ofthe generalized Dirac oscillator with interaction function f μ (x μ ) to be taken as the exponential-type potential can be mapped into the well-known hypergeometric equation and the analytical solutions can have been found.

Conclusion
In this work, the generalized Dirac oscillator has been studied in the presence of the gravitational fields of a cosmic string. The corresponding radial equation of generalized Dirac oscillator is obtained. In our generalized Dirac oscillator model, we take the interaction function f μ (x μ ) to be as cornell potential, Yukawa potential, generalized morse potential, Hulthé n-Type potential and singular potential respectively. By solving the corresponding wave equations the corresponding energy eigenvalues and the wave functions have been obtained and we have showed how the cosmic string, leads to modifications in the spectrum and wave function. Based on consideration that Dirac oscillator has been studied extensively in high energy physics, condensed matter physics, quantum Optics , mathematical physics and even in connection with Higgs symmetry it also makes sense to generalize the generalized Dirac oscillator to these fields.