SPIN AND PSEUDOSPIN SYMMETRIES IN RELATIVISTIC TRIGONOMETRIC PÖSCHL TELLER POTENTIAL WITH CENTRIFUGAL BARRIER

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1 International Journal of Modern Physics E Vol., No. 0) pages) c World Scientific Publishing Company DOI: 0.4/S SPIN AND PSEUDOSPIN SYMMETRIES IN RELATIVISTIC TRIGONOMETRIC PÖSCHL TELLER POTENTIAL WITH CENTRIFUGAL BARRIER M. HAMZAVI Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran majid.hamzavi@gmail.com S. M. IKHDAIR Physics Department, Near East University, 90 Nicosia, North Cyprus, Mersin 0, Turkey sikhdair@neu.edu.tr K.-E. THYLWE KTH-Mechanics, Royal Institute of Technology, S Stockholm, Sweden ket@mech.kth.se Received 8 August 0 Revised 4 September 0 Accepted October 0 Published 4 December 0 Approximate analytical solutions of the Dirac equation with the trigonometric Pöschl Teller tpt) potential are obtained for arbitrary spin-orbit quantum number κ using an approximation scheme to deal with the spin-orbit coupling terms κκ ± )r. In the presence of exact spin and pseudo-spin p-spin) symmetric limitation, the bound state energy eigenvalues and the corresponding two-component wave functions of the Dirac particle moving in the field of attractive and repulsive tpt potential are obtained using the parametric generalization of the Nikiforov Uvarov NU) method. The case of nonrelativistic limit is studied too. Keywords: Dirac equation; trigonometric Pöschl Teller tpt) potential; spin and p-spin symmetry; NU method; approximation schemes. PACS Numbers): Ge, Fd, Pm, 0.30.Gp,.90.+f Corresponding author

2 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe. Introduction Itiswellknownthattheexactenergyeigenvaluesoftheboundstatesplayanimportant role in quantum mechanics. In particular, the Dirac equation, which describes the motion ofaspin-/particle,has been usedin solvingmanyproblemsofnuclear and high-energy physics. The spin and the p-spin symmetries of the Dirac Hamiltonian had been discovered many years ago; however, these symmetries have recently been recognized empirically in nuclear and hadronic spectroscopes. Within the framework of Dirac equation, p-spin symmetry is used to feature the deformed nuclei and the super deformation to establish an effective shell model, 4 whereas spin symmetry is relevant for mesons. 5 The spin symmetry occurs when the scalar potential Sr) is nearly equal to the vector potential Vr) or equivalently Sr) Vr) and p-spin symmetry occurs when Sr) Vr). 6,7 The p-spin symmetry refers to a quasi-degeneracy of single nucleon doublets with nonrelativistic quantum number n,l, j = l+/) and n, l+, j = l+3/), where n, l and j are single nucleon radial, orbital and total angular quantum numbers, respectively. 8,9 The total angular momentum is given by j = l+ s, where l = l+ pseudo-angular momentum and s is p-spin angular momentum. 0, Liang et al. investigated the symmetries of the Dirac Hamiltonian and their breaking in realistic nuclei in the framework of perturbation theory. Guo 3 used the similarity renormalization group to transform the spherical Dirac operator into a diagonal form and then the upper lower) diagonal element became an operator describing Dirac anti-)particle, which holds the form of the Schrödinger-like operator with the singularity disappearing in every component. Chen and Guo 4 investigated the evolution toward the nonrelativistic limit from the solutions of the Dirac equation by a continuous transformation of the Compton wavelength λ. Lu et al. 5 recently showed that the p-spin symmetry in single particle resonant states in nuclei is conserved when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign. The tpt potential has been proposed for the first time by Pöschl and Teller 6 in 933 to describe the diatomic molecular vibration. Chen 7 and Zhang et al. 8 have studied the relativistic bound state solutions for the tpt potential and hyperbolical PT Second PT) potential, respectively. Liu et al. 9 studied the tpt potential within the framework of the Dirac theory. Recently, Candemir 0 investigated the analytical s-wave solutions of Dirac equation for tpt potential under the p-spin symmetry condition. Very recently, Hamzavi and Rajabi studied the exact s- wave solution l = 0) of the Schrödinger equation for the vibrational tpt potential. The tpt takes the form: V V tpt r) = sin αr) + V cos, 0 < αr < π/, ) αr) where the parameters V and V describe the property of the potential while the parameter α is related to the range of this potential. 9 We find out that this potential has a minimum value at r 0 = α tan V 4 V ). For the case when V = V, the minimum value is at r 0 = π 4α 0, ) for α > 0. The second derivative which

3 Approximate Solutions of the Dirac Equation determines the force constants at r = r 0 is given by d V dr for any α value and thus = 8α V + V V ) [ )], ) r=r0 cos tan 4 V V Vr 0 ) = V V +V [ cos tan 4 V V )], 3) which means that Vr) at r = r 0 has a relative minimum for α > 0. When V = V = V then minimum value is Vr 0 ) = 4V and d V dr r=r0 = 3α V. In Figs. a) and b), we draw the tpt potential ) for parameter values V = 5.0 fm, V = 3.0 fm, α = 0.0 fm and α = 0.30 fm. Here the potential has a minimum value at r 0 = 0.707π/α. The curve is nodeless in αr 0,π/). For example, with α = 0.30 fm,r 0 =.8303 fm and minimum potential Vr 0 =.8303 fm) = fm. It is worthy to note that in the limiting case when α 0, the tpt potential can be reduced to the Kratzer potential,3 Vr) = D e r re r ) +η, where r e is the equilibrium intermolecular separation and D e is the dissociation energy between diatomic molecules. In our case, D e = V, η = V and r e = /α. In the case of η = 0, it reduces to the molecular potential which is called the modified Kratzer potential proposed by Simons et al. 4 and Molski and Konarski. 5 In the case of η = D e, this potential turns into the Kratzer potential, which includes an attractive Coulomb potential and a repulsive inverse square potential introduced by Kratzer in The aim of this present work is to extend our previous work 0 to the relativistic case and κ ± rotational case). We introduce a convenient approximation scheme to deal with the strong singular centrifugal term. The ansatz of this approximation possesses the same form of the potential and is singular at the origin as the centrifugal term r. We want to solve the Dirac equation with flexible parameters tpt potential model. However, the Dirac-tPT problem can no longer be solved κ ± in a closed form due to the existence of spin-orbit coupling term κκ±)r and it is necessary to resort to approximation methods. Therefore, we use an approximation scheme to deal with this term and solve approximately the Dirac equation with the tpt potential for arbitrary spin-orbit quantum number κ. In the presence of spin and p-spin symmetric limitation, we obtain the approximate relativistic bound state solutions including the energy eigenvalue equations and the corresponding unnormalized upper- and lower-spinor components of the wave functions using the concepts of parametric generalization of the NU method, 7 since the relativistic corrections are not neglected

4 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe a) b) Fig.. a) A plot of the tpt potential for α = 0. fm. b) A plot of the tpt potential for α = 0.8 fm

5 Approximate Solutions of the Dirac Equation α =0.00 Vr)=α / sin α r) Vr)=α d 0 +α / sin α r) Vr)=/r Vr) r Fig.. Color online) The centrifugal term /r green line) and its approximations Eq. 8). Over the past years, the Nikiforov Uvarov NU) method 8 has shown to be a powerful tool in solving the second-order differential equation. It was applied successfully to a large number of potential models This method has also been used to solve the spinless spin-0) Schrödinger and Klein Gordon KG) 4 46 equations and also relativistic spin-/ Dirac equation 47 5 with different potential models. The structure of the paper is as follows. In Sec., in the context of spin and p-spin symmetry, we briefly introduce the Dirac equation with scalar and vector tpt potentials for arbitrary spin-orbit quantum number κ. The parametric generalization of the NU method is displayed in Appendix A. In the presence of the spin and p-spin symmetry, the approximate energy eigenvalue equations and corresponding two-component wave functions of the Dirac-tPT problem are obtained. The nonrelativistic limit of the problem is discussed in this section too. Finally, our final concluding remarks are given in Sec. 3.. Bound State Solutions The Dirac equation for fermionic massive spin-/ particles moving in the field of an attractive scalar Sr) and a repulsive vector Vr) potential in units = c = )

6 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe is [α p+βm +Sr))]ψr) = [E Vr)]ψr), 4) where E is the relativistic energy of the system, p = i is the three-dimensional 3D) momentum operator and M is the mass of the fermionic particle. α and β are the 4 4 usual Dirac matrices. 53 One may closely follow the procedure described in Eqs. 7) 9) of Ref. 54 to obtain, [ d dr κκ+) r ] F nκ r)+ d r) dr d M +E nκ r) dr + κ ) F nκ r) r = [M +E nκ r))m E nκ +Σr))]F nκ r), r 0, ), 5) [ d dr κκ ) r ] G nκ r)+ dσr) dr d M E nκ +Σr) dr κ ) G nκ r) r = [M +E nκ r))m E nκ +Σr))]G nκ r), r 0, ), 6) where κκ ) = l l+) and κκ+) = ll+). The spin-orbit quantum number κ is related to the orbital quantum numbers l and l for spin symmetry and p-spin symmetric models, respectively, as l+) = κ = j + ) s /,p 3/,...), j = l+, aligned spin κ < 0), +l = + j + ) p /,d 3/,...), j = l, unaligned spin κ > 0). Further, κ in the quasi-degenerate doublet structure can be expressed in terms of s = / and l, the p-spin and pseudo-orbital angular momentum, respectively, as l = j + ) s /,p 3/,...), j = l, aligned p-spin κ < 0), κ = + l+) = + j + ) d 3/,f 5/,...), j = l+, unaligned p-spin κ > 0), where κ = ±,±,... For example, the states s /,0d 3/ ) and p 3/,0f 5/ ) can be considered as p-spin doublets... Spin symmetric limit In the spin symmetric limitation, d r) dr Eq. 5) with Σr) = V tpt r), becomes [ d dr κκ+) r γ = 0 or r) = C s = constant, 7,55 58 then V sin αr) + V cos αr) ) β ] F nκ r) = 0, 7a) γ = M +E nκ C s and β = M E nκ )M +E nκ C s ), 7b)

7 Approximate Solutions of the Dirac Equation where κ = l and κ = l for κ < 0 and κ > 0, respectively. The Schrödinger-like equation 7a) that results from the Dirac equation is a second-order differential equation containing a spin-orbit centrifugal term κκ + )r which has a strong singularity at r = 0, and needs to be treated very carefully while performing the approximation. Equation 7a) has an exact rigorous solution only for the states with κ = because of the existence of the centrifugal term κκ+)/r. However, when this term is taken into account, the corresponding radial Dirac equation can no longer be solved in a closed form and it is necessary to resort to approximate methods. Over the last few decades several schemes have been used to calculate the energy spectrum. The main idea of these schemes relies on using different approximations of the spin-orbit centrifugal coupling term κκ + )/r. So we need to perform a new approximation for the spin-orbit term as a function of the tpt potential parameters. Therefore, we resort to use an appropriate approximation scheme to deal with the centrifugal potential term as r = lim α 0 α d 0 + sin αr) ), 0 < αr, 8) where d 0 = / is a dimensionless shifting parameter. The approximation 8) is done on the basis that sinz) = z z 3 /3! + z 5 /5! z 7 /7! +, and in the limit when z 0, sinz) z. To show the validity and accuracy of our choice to the approximation scheme 8), we plot the centrifugal potential term /r and its approximations:α /sin αr) and α d 0 +/sin αrsin αr)) in Fig.. As illustrated, the three curves coincide together and show how accurate is this replacement. One of us has treated this problem in his recent work see Ref. 59). Thus, employing such an approximation scheme, we can then write Eq. 7a) as [ d dr κκ+)α F nκ r) = 0. d 0 + sin αr) ) γ V sin αr) + V cos αr) ) β ] 9) Followed by making a new change of variables sr) = sin αr), this allows us to decompose the spin-symmetric Dirac equation 9) into the Schrödinger-type equation satisfying the upper-spinor component F n,κ s), d ds + s d s s) ds s s) [ As +Bs C] F n,κs) = 0, A = 4α [κκ+)α d 0 +β ], B = 4α [κκ+)α d 0 )+γv V ) β ], C = 4α [κκ+)α +γv ], 0)

8 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe Table. The specific values of the parametric constants for the spin symmetric Dirac-tPT problem. Constant Analytic value c 4 4 c 5 c A) c 7 +4B) 4 c C) c 9 A B +C + 6 c 0 +6C c A B +C + 6 c C) c A B +C + 6 where F nκ r) F n,κ s) has been used. If the above equation is compared with A.), we can obtain the specific values for constants c i i =,,3) as c =, c = and c 3 =. InordertoobtaintheboundstatesolutionsofEq.9),itisnecessarytocalculatethe remaining parametric constants, i.e., c i i = 4,5,...,3) by means of the relation A.5). Their specific values are displayed in Table for the relativistic tpt potential model. Further, using these constants along with A.0), we can readily obtain the energy eigenvalue equation for the Dirac-tPT problem as or equivalently n++ n+ + A B +C V M +E nκ C s ) α + C + 6 ) = A, ) κ+) + 4V M +E nκ C s ) α = α E nκ M)M +E nκ C s ) κκ+)d 0. ) )

9 Approximate Solutions of the Dirac Equation To show the procedure of determining the energy eigenvalues from Eq. ), we take a set of physical parameter values, M = 0 fm, V = 5.0 fm, V = 3.0 fm, C s = 0 fm and α = 0.8, 0.6, 0.4, 0., 0.04, In Tables and 3, we present the energy spectrum for the spin symmetric case. Obviously, the pairs np /,np 3/ ), nd 3/,nd 5/ ), nf 5/,nf 7/ ), ng 7/,ng 9 ), and soonaredegenerate states. Thus, each pair is considered as spin doublet and has positive energy. 54 Further, when potential range parameter α approaches zero, the energy eigenvalues approaches a constant. From Eq. ) we find that this constant is M +V +V + V V, i.e. lim α 0 E n,κ = M + V + V + V V, where it can be seen from Tables and 3, too. In addition, to show the accuracyofour approximationscheme, we have calculated the exact numerical energy eigenvalues of Eq. 7a) without making approximation to the centrifugal term by using the amplitude phase AP) method As shown in Tables and 3, our approximate energies obtained via the NU method is highly accurate 0.000%) if compared with the exact ones obtained via AP in the low screening regime when the screening parameter values: α = 0.0 fm and 0.04 fm. Also the approximate energies for the screening parameter values α = 0. fm and 0.4 fm are also accurate %). This means that the present approximation works well for the low values of α. On the other hand, in order to establish the upper-spinor component of the wave functions F n,κ r), namely, Eq. 7a), the relations A.) A.4) are used. First, we find the first part of the wave function as φs) = s C) s) 4 + A B+C+ 6. 3) Second, we calculate the weight function as ρs) = s +6C s) A B+C+ 6, 4) which gives the second part of the wave function as +6C, A B+C+ 6 ) y n s) = P n s), 5) where P n a,b) y) are the orthogonal Jacobi polynomials. Finally, the upper spinor component for arbitrary κ can be found through the relation A.4) F nκ s)=n nκ s C) s) 4 + A B+C+ 6 P or +6C, A B+C+ 6 ) n s) 6) F nκ r) = N nκ sinαr)) +ηκ) cosαr)) +δ) P ηκ, δ) n cosαr)), 7) where η κ = + 4 α [κκ+)α +γv ], δ = + 4γV α 8)

10 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe Table. The spin symmetric bound state energy eigenvalues fm ) of the tpt potential for several values of n and κ with M = 0.0, V = 5.0, V = 3.0, cs = 0 and d0 = /. α = 0.8 α = 0.8 α = 0.4 α = 0.4 α = 0. α = 0. α = 0.04 α = 0.04 α = 0.0 α = 0.0 l n, κ l, j = l ± /) NU AP NU AP NU AP NU AP NU AP,, p 3/,p / , 3, d 5/,d 3/ , 4,3 f 7/,f 5/ , 5,4 g 9/,g 7/ ,, p 3/,p / , 3, d 5/,d 3/ , 4,3 f 7/,f 5/ , 5,4 g 9/,g 7/ Table 3. The spin symmetric bound state energy eigenvalues fm ) of the tpt potential for several values of n and κ with d0 = 0. α = 0.8 α = 0.8 α = 0.4 α = 0.4 α = 0. α = 0. α = 0.04 α = 0.04 α = 0.0 α = 0.0 l n, κ l, j = l ± /) NU AP NU AP NU AP NU AP NU AP,, p 3/, p / , 3, d 5/, d 3/ , 4,3 f 7/, f 5/ , 5,4 g 9/, g 7/ ,, p 3/, p / , 3, d 5/, d 3/ , 4,3 f 7/, f 5/ , 5,4 g 9/, g 7/

11 Approximate Solutions of the Dirac Equation and N nκ is the normalization constant. Further, the lower-spinor component of the wave function can be calculated by using d G nκ r) = M +E nκ C s dr + κ ) F nκ r), 9) r where E M + C s and in the presence of the exact spin symmetry C s = 0), only positive energy states do exist... p-spin symmetric limit Ginocchio showed that there is p-spin symmetry in case when the relationship between the vector potential and the scalar potential is given by Vr) = Sr). 7 Further, Meng et al. showed that if d[vr)+sr)] dr = dσr) dr = 0, then Σr) = C ps = constant, for which the p-spin symmetry is exact in the Dirac equation Thus, choosing the r) as tpt potential, Eq. 6) under this symmetry becomes [ d dr κκ ) ] V r γ sin αr) + V ) cos αr) β G nκ r) = 0, 0a) γ = E nκ M C ps and β = M +E nκ )M E nκ +C ps ), 0b) where κ = l and κ = l + for κ < 0 and κ > 0, respectively. Employing the new approximation for the spin-orbit pseudo-centrifugal term, κκ )/r, i.e., Eq. 8), the p-spin Dirac equation 0a) can be written as [ d dr κκ )α d 0 + sin αr) ) γ V sin αr) + V cos αr) ) β ] G nκ r) = 0. ) To avoid repetition, the negative energy solution of Eq. ), the p-spin symmetric case can be readily obtained directly via the spin symmetric solution throughout the following parametric mappings: F nκ r) G nκ r), κ κ, Vr) Vr) i.e. V V,V V ), E nκ E nκ, C s C ps. Following the previous procedure, one can obtain the p-spin symmetric energy equation as n++ + 4V E nκ M C ps ) α + ) κ ) + 4V E nκ M C ps ) α ) = α M +E nκ)m E nκ +C ps )+κκ )d 0. 3) Furthermore, the lower-spinor component of the wave functions is found as G nκ r) = Ñn,κsinαr)) + ηκ) cosαr)) + δ) P ηκ, δ) n cosαr)), 4)

12 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe Table 4. The bound state energy eigenvalues in unit of fm of the p-spin symmetry tpt potential for several values of n and κ with d0 = /. α = 0.8 α = 0.8 α = 0.4 α = 0.4 α = 0. α = 0. α = 0.04 α = 0.04 α = 0.0 α = 0.0 l n,κ l,j) NU AP NU AP NU AP NU AP NU AP,, s /, 0d 3/ ,,3 p 3/, 0f 5/ , 3,4 d 5/, 0g 7/ , 4,5 f 7/, 0h 9/ ,, s /, d 3/ ,,3 p 3/, f 5/ , 3,4 d 5/, g 7/ , 4,5 f 7/, h 9/ Table 5. The p-spin symmetric bound state energy eigenvalues fm ) of the tpt potential for several values of n and κ with d0 = 0. α = 0.8 α = 0.8 α = 0.4 α = 0.4 α = 0. α = 0. α = 0.04 α = 0.04 α = 0.0 α = 0.0 l n,κ l,j) NU AP NU AP NU AP NU AP NU AP,, s /, 0d 3/ ,,3 p 3/, 0f 5/ , 3,4 d 5/, 0g 7/ , 4,5 f 7/, 0h 9/ ,, s /, d 3/ ,,3 p 3/, f / , 3,4 d 5/, g 7/ , 4,5 f 7/, h 9/

13 Approximate Solutions of the Dirac Equation with η κ = κ ) + 4V E nκ M C ps ) α, δ = + 4V E nκ M C ps ) α, 5) where Ñnκ is the normalization constant. In Tables 4 and 5, we give the numerical results for the p-spin symmetric case. In this case, we take the set of parameter values, M = 0 fm, V = 5.0 fm, V = 3.0 fm, c ps = 0 fm and α = 0.8, 0.4, 0., 0.04, 0.0. We observe the degeneracy in the following doublets s /,0d 3/ ), p 3/,0f 5/ ), d 5/,0g 7/ ), f 7/,0h 9/ ), and so on. Thus, each pair is considered as p-spin doublet and has negative energy. 59,64 Furthermore, one can also see the exact numerical energy eigenvalues obtained via AP method to test the accuracy of the present approximate solutions found by using the NU method. As shown in Tables 4 and 5, the accuracy ofourresultsishigh whencomparedwith theexactonesin the lowscreeningregime small values of α). On the other hand, the upper-spinor component of the Dirac wave function can be calculated by d F nκ r) = M E nκ +C ps dr κ ) G nκ r), 6) r where E M +C ps and in the presence of the exact p-spin symmetry C ps = 0), only negative energy states do exist..3. The nonrelativistic limiting case In this section, we study the energy eigenvalue equation ) and upper-spinor component of wave function7) of the Dirac-tPT problem under the nonrelativistic limits: C s = 0,κ l, E nκ M E nl and M + E nκ m. Thereby, we obtain the energy equation of the Schrödinger equation with any arbitrary orbital state for the tpt potential as E nl = α ll+)d 0 m [ + α n+ m + l+) 4 + 8mV )] α + + 8mV α. 7) In the limit when α 0, the vibration-rotation energy formula 7) reduces into a constant value: lim E nl = V + V ). 8) α 0 Further, there is no loss of generality if d 0 = 0, then Eq. 7) becomes [ E nl = α n+ m + l+) 4 + 8mV )] α + + 8mV α, 9)

14 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe Table 6. The bound state energy levels Enl of the Schrödinger equation for the tpt potential. Enl M = 0.0 fm, V = 5 fm, V = 3 fm Ref. 5) and d0 = State n,l) α =. α = 0.8 α = 0.4 α = 0. α = 0.0 α = 0.00 s Ref. ) s Ref. ) p s Ref. ) p d s Ref. ) p d f Table 7. The bound state energy levels Enl of the Schrödinger equation for the tpt potential. Enl M = 0 fm, V = 5 fm, V = 3 fm Ref. 4) and d0 = 0 State n,l) α =. α = 0.8 α = 0.4 α = 0. α = 0.0 α = 0.00 s Ref. ) s Ref. ) p s Ref. ) p d s Ref. ) p d f

15 Approximate Solutions of the Dirac Equation where n = 0,,,... and l = 0,,,... are the vibration and rotation quantum numbers, respectively. To obtain a numerical energy eigenvalues for the present potential model, we take the following set of parameter values; namely, M = 0 fm, V = 5.0 fm, V = 3.0 fm and α =., 0.8, 0.4, 0., 0.0, ,,65 As seen fromtables5and 6,inthe limit whenpotential rangeparameterαapproacheszero, the energy eigenvalues approaches a constant value given by Eq. 8). In Tables 6 and 7, we take d 0 = / and d 0 = 0, respectively. Also we can get the radial wave functions of the Schrödinger equation with tpt potential as R n,l s) = s 4 [ ) + l+) + 8mV α ] s) mV α ) l+) + 8mV α, + 8mV α Pn s). 30) Inserting s = sin αr) in the above equation, we can obtain R n,l r) = N nl sinαr)) +ηl)/ cosαr)) +δ)/ P η l/,δ/) n cosαr)), 3a) η l = l+) + 8mV α, δ = + 8mV α 3b) where N nl is a normalization factor to be calculated from the normalization conditions. To conclude, we need to mention that one of the applications of the nonrelativistic tpt model is in molecular physics, to calculate the energy spectrum of the diatomic molecules Concluding Remarks In this work, we have investigated the bound state solutions of the Dirac equation with trigonometric Pöschl Teller potential for any spin-orbit quantum number κ. By making an appropriate approximation to deal with the spin-orbit centrifugal pseudo-centrifugal) coupling term, we have obtained the approximate energy eigenvalue equation and the unnormalized two components of the radial wave functions expressed in terms of the Jacobi polynomials using the NU method. Also, we obtained the exact numerical energy eigenvalues via AP method to test the accuracy of the present solutions. Furthermore, we have obtained the nonrelativistic solutions for the rotation-vibration energy eigenvalues and corresponding radial wave function of the spin-0 particle moving under the influence of the tpt field. Acknowledgments We thank the kind referees for the positive suggestions and critics which have greatly improved the present paper. M. Hamzavi thanks his host institution, KTH- Mechanics, Royal Institute of Technology, S Stockholm, Sweden

16 M. Hamzavi, S. M. Ikhdair & K.-E. Thylwe Appendix A. Parametric Generalization of the NU Method The NU method is used to solve second-order differential equations with an appropriate coordinate transformation s = sr) Ref. 4) ψ n τs) σs) s)+ σs) ψ n s)+ σ s) ψ ns) = 0, A.) where σs) and σs) are polynomials, at most of second degree, and τs) is a firstdegree polynomial. To make the application of the NU method simpler and direct without need to check the validity of solution. We present a shortcut for the method. So, at first we write the general form of the Schrödinger-like equation A.) in a more general form applicable to any potential as follows 7 : ψ n s)+ c c s s c 3 s) satisfying the wave functions ) ψ n s)+ ξ s +ξ s ξ 3 s c 3 s) ψ n s) = ϕs)y n s). ) ψ n s) = 0, A.) A.3) ComparingA.) with its counterparta.), we obtain the following identifications: τs) = c c s, σs) = s c 3 s), σs) = ξ s +ξ s ξ 3. Following the NU method, 4 we obtain the following 7 : i) the relevant constant: c 4 = c ), c 5 = c c 3 ), c 6 = c 5 +A, c 7 = c 4 c 5 B, c 8 = c 4 +C, c 9 = c 3 c 7 +c 3 c 8 )+c 6, c 0 = c +c 4 + c 8 >, c = c c 4 + c 3 c9 >, c 3 0, A.4) A.5) c = c 4 + c 8 > 0, c 3 = c 4 + c 3 c 9 c 5 ) > 0, c 3 0. ii) The essential polynomial functions: πs) = c 4 +c 5 s [ c 9 +c 3 c8 )s c 8 ], k = c 7 +c 3 c 8 ) c 8 c 9, τs) = c +c 4 c c 5 )s [ c 9 +c 3 c8 )s c 8 ], A.6) A.7) A.8) τ s) = c 3 c 9 +c 3 c8 ) < 0. A.9) iii) The energy equation: c c 3 )n+c 3 n n+)c 5 +n+) c 9 +c 3 c8 ) +c 7 +c 3 c 8 + c 8 c 9 = 0. A.0)

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