Analytic solutions of the Schr\"odinger equation with non-central generalized inverse quadratic Yukawa potential

This study presents the solutions of Schr\"odinger equation for the Non-Central Generalized Inverse Quadratic Yukawa Potential within the framework of Nikiforov-Uvarov. The radial and angular part of the Schr\"odinger equation are obtained using the method of variable separation. More so, the bound states energy eigenvalues and corresponding eigenfunctions are obtained analytically. Numerical results were obtained for the Generalized Inverse Quadratic Yukawa Potential for comparison sake. It was found out that our results agree with existing literature.


INTRODUCTION
Physicists over the years have developed strong interest in searching for the solution of the Schrödinger equation with some potentials [1][2][3][4][5][6][7].This is because, finding the analytical solution of the Schrödinger equation is extremely crucial in nonrelativistic quantum mechanics and the eigenfunction contains all the necessary information required to describe a quantum system under consideration.
This potential is a superposition of the inverse quadratic Yukawa (IQY) [32] and the Yukawa potential [50].It is asymptotic to a finite value as  → ∞ and becomes infinite at  = 0. [51] solved this potential within the framework of the proper quantization rule and eigenfunction was obtained via the Formula Method [52].The generalized inverse quadratic Yukawa potential model is of the form [51] [48] Compared the behaviour of the Yukawa-type potential with the Yukawa potential and the IQY potential for screening parameter values, it was noted that differences do not exist between these three potentials.
The Ring shaped potential have a wide range of applications in quantum chemistry and nuclear physics [53].They have very important role in describing ring-shaped molecules like benzene and the interactions between deformed pair of nuclei [54][55].They have also been used in demonstrating some of the pseudospin symmetry in nuclei physics [27].The exact results can be used in accounting for some axial symmetric system in quantum chemistry.[26] proposed a non-central potential as; Where  = ℋ =  1 and ℧ = 2 1 The Generalized inverse quadratic Yukawa potential reduces to a constant potential when  = ℧ = 0.
The organization of the work is as follows: In the next section, we give a review of the NU method.In Sect.3, this method is applied to obtain the bound state solutions with Non-Central Generalized Inverse Quadratic Yukawa Potential.In Sect.4, we obtain numerical results while in Sect.5, we give a brief concluding remark.

REVIEW OF NIKIFOROV-UVAROV METHOD
The Nikiforov-Uvarov (NU) method is based on solving the hypergeometric-type second-order differential equations by means of the special orthogonal functions [56][57].The main equation which is closely associated with the method is given in the following form [58]; Where () and  ̃() are polynomials at most second-degree, () is a first-degree polynomial and () is a function of the hypergeometric-type.
The exact solution of Eq. ( 4) can be obtained by using the transformation This transformation reduces Eq. ( 4) into a hypergeometric-type equation of the form The function () can be defined as the logarithm derivative where with () being at most a first-degree polynomial.The second () being   () in Eq. ( 5), is the hypergeometric function with its polynomial solution given by Rodrigues relation Here,   is the normalization constant and () is the weight function which must satisfy the condition It should be noted that the derivative of with respect to should be negative.The eigenfunctions and eigenvalues can be obtained using the definition of the following function and parameter , respectively: where The value of can be obtained by setting the discriminant of the square root in Eq. ( 12) equal to zero.As such, the new eigenvalue equation can be given as

SEPARATION OF VARIABLES FOR THE SCHRODINGER EQUATION
In spherical coordinates (, , ), the Schrodinger equation with potentials (, ), respectively, can be written as follows [26]: ∇ 2 (, , ) + (, )(, , ) = (, , ) where  is the non-relativistic energy of the system,  denotes the rest mass of the particle and ℏ is the planck constant.The Schrodinger equation with potential is given by [54]; Substituting Eq. (3) into Eq.(16), we have )) − ] (, , ) = 0 Substituting Eq. ( 17) into Eq.( 18) and using the standard procedure of separating variables, we obtain the following differential equations: where  2 and Λ are separation constants, which are real and dimensionless.The solution of Eq. ( 21) is periodic and for bound state Φ() satisfies the periodic boundary condition Φ( + 2) and its solutions become,

Solutions of the radial Schrodinger equation for Non-Central Generalized Inverse
Quadratic Yukawa Potential The radial part of the Schrödinger equation for this potential can be solved exactly for  = 0 (s-wave) but cannot be solved for this potential for  ≠ 0. To obtain the solution for  ≠ 0, we employ the approximation scheme proposed by Greene and Aldrich [59] to deal with the centrifugal term, which is given as; It is noted that for a short-range potential, the relation (eqs.24 ) is a good approximation to , as proposed by Greene and Aldrich [59,[60][61].The implies that Eq. ( 24) is not a good approximation to the centrifugal barrier when the screening parameter  becomes large.Thus, the approximation is valid when  << 1. Substituting the approximation (Eq.24) into Eq.( 23), we obtain an equation of the form; Eq. ( 25) can be simplified into the form and introducing the following dimensionless abbreviations Using a transformation  =  −  so as to enable us apply the NU method as a solution of the hypergeometric type We obtain the differential equation Comparing Eq. ( 29) and Eq. ( 4), we have the following parameters To find the constant the discriminant of the expression under the square root of Eq. (31) should be equal to zero.As such, we have that Substituting Eq. ( 33) into Eq.( 31) yields where From the knowledge of NU method, we choose the expression which the function has a negative derivative.This is given by with being obtained as Referring to Eq. ( 13), we define the constant as Taking the derivative of () from Eq.(37), we have; and () from Eq.( 30), we have; Substituting Eq. ( 39) into Eq.( 40), we have Comparing Eq. ( 38) and ( 41), and carrying out some algebraic manipulation.We have; Substituting Eqs.(17) and Eq. ( 32) into Eq.( 31) yields the energy eigenvalue equation of the Hellman potential in the form The corresponding wave functions can be evaluated by substituting from Eq.
(34) and Eq. ( 30) respectively into Eq.( 7) and solving the first order differential equation.This gives The weight function from Eq. ( 10) can be obtained as From the Rodrigues relation of Eq. ( 9), we obtain where is the Jacobi Polynomial.
Substituting Α()    () from Eq. ( 44) and Eq. ( 46) respectively into Eq.( 5), we obtain the wave function in terms of hyper-geometric polynomial as where Ω , is a normalization constant,  = Using the normalization condition, we obtain the normalization constant as follows [60]: Substituting Eq. ( 47) into Eq.( 50), we have ( ) ( ) s and s To find the constant the discriminant of the expression under the square root of Eq. ( 58) should be equal to zero.As such, we have that Substituting Eq. ( 60) into Eq.( 58) yields From the knowledge of NU method, we choose the expression which the function has a negative derivative.This is given by with being obtained as Referring to Eq. ( 13), we define the constant as Taking the derivative of () from Eq.(63), we have; The corresponding wave functions can be evaluated by substituting from Eq.
(57) and Eq. ( 61) respectively into Eq.( 7) and solving the first order differential equation.This gives , k Our results shown in Table 1 are in good agreement with the result of Ref. [51] who solved the Schrodinger Equation with this potential using the Proper Quantization rule within the framework of the Pekeris Approximation scheme.

CONCLUSIONS
We have obtained exact bound states solutions of three dimensional three dimensional Schrödinger equation for the Non-Central Generalized Inverse Quadratic Yukawa Potential within the framework of Nikiforov-Uvarov method.We have solved the corresponding eigenvalues and eigenfunctions of the radial and angular parts of the Schrödinger equation.More so, when the ring shaped like term vanishes, i.e.  =  =  = 0 then the results are in good agreement with Ref. [51].Different ring shaped like potentials can be obtained from this new proposed Non central like potential.Finally, our results can find many applications in nuclear physics and quantum chemistry such as cyclic benzene.

Table 1 .
The bound state energy levels (in units of fm−1) of the GIQYP for various values of , ℓ and for ℏ =  = 1.