Solutions of the D-dimensional Schrodinger equation with the hyperbolic Poschl Teller potential plus modified ring shaped term

In this paper, we solve the D-dimensional Schr\"odinger equation with hyperbolic Poschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate. We used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the angular part are solved using the NU approach as well.


Introduction
The non-central potentials in recent times have been an active field of research in physics and quantum chemistry [1][2][3]. For instance, the occurrence of accidental degeneracy and hidden symmetry in the non-central potentials and its application in quantum chemistry and nuclear physics are used to describe ring-shaped molecules like benzene and the interaction between deformed pair of nuclei [4][5]. It is known that these accidental degeneracy occurring in the ring shaped was explain by constructing an SU (2) algebra [6]. Owing to these applications many authors have investigated a number of real physical problems on non-spherical oscillator [7], ring-shaped oscillator (RSO) [8] and ring shaped non-spherical oscillator [9]. Berdemir [10] had shown that either Coulomb or harmonic oscillator will give a better approximation for understanding the spectroscopy and structure of diatomic molecules in the ground electronic state. Other applications of the ring shaped potential can be found in ring shaped organic molecules like cyclic polyenes and benzene [11][12].

D-Dimensional Schrodinger equation in Hyperspherical Coordinates
The D-dimensional Schrodinger equation is given below [34][35]     where  is the effective mass of two interacting particles,  is Planck's constant, E is the energy eigenvalue, U is the potential energy function, 1 2 1 ( , , ,...., is the angular position vector written in terms of Hyperspherical coordinates [36][37], and 2 D  is the Ddimensional Laplacian operator given in Appendix B. The solvable potentials which allows separation of variable in (2.1) must be of the form: . The separable wavefunction take the following form: where we assumed the following conditions: are polynomials of degrees 2 and 1, respectively. In this case, Eq. (3.5) will be used to obtain the energy spectrum formula of the quantum mechanical system. We should point out here that the polynomial solutions to Eq. (3.4) for 0    and 0   on the boundaries of the finite space (the latter case is omitted for infinite space), are the classical orthogonal polynomials. It is well known that each set of polynomials is associated with a weight function   function must be bounded on the domain of the system and must satisfy       . This weight function will be used to construct the Rodrigues formula for these polynomials which reads: where n B is just a constant obtained by the normalization conditions, and n =0, 1, 2,….

The Solutions of the D-dimensional Radial equation
We use the NU method to solve equation (2.4), in the presence of our potential, . Equation (4.1) cannot be solved analytically due to the centrifugal term 2 1 r . Different authors used different approximation techniques to allow an approximate analytical solution of (4.1) and these methods rely on Taylor expansion of the centrifugal potential in terms of the other components of the potential of interest [39]. In this work, we use the following approximation obtained by Taylor expansion [40], 2) The advantage of this approximation it is valid not only for 1 r   but also for The choice of k that makes (4.4) a polynomial of first degree must satisfy 2 where we will pick the negative part in (4.5) that makes 0 where conditions for bound states becomes We will only take the (-) sign in (4.11) as explained below. The s-wave spectrum formula in three dimensions is the only exact solution which is obtained by setting 0 D   in (4.11). However, for other higher states, the above solution is acceptable with high accuracy as far as the condition 0 2 r    is satisfied.
To calculate the integral in (4.13), we will use the following very useful integral formula [42]     , , ; , ; F a b c d e f is the generalized hypergeometric function [42]. By direct comparison between (4.13) and (4.14) we get 1/ n n    , where n  is given below where 6 7 a c c    and The only issue that is left for discussion in this section is that the solutions of the radial wave equation where N denotes the maximum number in which we get bound states, are not orthogonal! But they are normalized as we discussed above. We know that Hermitian operators with distinct eigenvalues must have orthogonal eigenvectors [43]. To solve this problem one must use the method of Gram-Schmidt (GS) to obtain an orthonormal set     0 N n n r   by linear combinations [44]. The latter set will be the solutions of the radial wave equation. The process is a bit lengthy and we will not be able to do it here. However, we encourage the interested reader to do these calculations by referring to the process of GS.

Solutions of the angular equations
It is well-known from literature that solutions of (7) are written in terms of Jacobi polynomials as follows [42]           , and the parameter k defined in (3.3.f) must satisfy The latter constraint will be used later to obtain the eigenvalues of Eq.
In the next section, we will consider different examples and try to obtain the unknown parameters for each case.

Results and Discussions
In this section, we will discuss different examples that are considered as special cases of the potential in (1.1).
As a first example, we consider the case when 0,  (2)The next special case of our potential model is considered when we choose the ring shaped parameters , 0       , which corresponds to the following potential ( ,

Conclusions
In this paper, we have obtained analytically the solutions of the D-dimensional Schrödinger potential with hyperbolic Poschl Teller potential plus a generalized ringshaped term. We employed NU and trial function methods to solve the radial and angular part of the Schrödinger equation respectively. This result is new and has never been