# Approximate Solutions to the Klein-Fock-Gordon Equation for the Sum of Coulomb and Ring-Shaped-Like Potentials

Authors: Sh. M. Nagiyev; A. I. Ahmadov; V. A. Tarverdiyeva
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with massM, described by the Klein-Fock-Gordon equation with equal scalar Sr→ and vector Vr→ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at E<Mc2 and a continuous at E>Mc2 energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group SU1,1 for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra, and group generators in the limit c⟶∞ go over into the corresponding expressions for the nonrelativistic problem.


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## Body

## 1. Introduction

Nonrelativistic Schrödinger and relativistic Dirac, Klein-Fock-Gordon (KFG), and finite-difference equations describe the systems in the nuclear physics, elementary particle physics, and atomic and molecular physics [1–6]. Examples to the commonly used potentials in these equations are the Coulomb potential and the harmonic oscillator potential, as well as their various varieties. Other types of interaction potentials included were the Kratzer [7], Morse [8], Eckart [9], Manning–Rosen [10], Pöschl-Teller [11], Hulthèn [12], Wood-Saxon [13], Makarov [14], Hartmann [15], and Hautot [16] potentials. Exact solvability and the application range of these potentials are examples to the main properties of them. Several approaches were developed to study the quantum systems either relativistic or nonrelativistic with nonzero angular momentum [17–23].The numerous interesting works devoted to the study of the KFG equation with different potentials. The s-wave KFG equation with the vector Hulthèn-type potential is one of the examples which was investigated by the use of the standard methods. Afterwards, the further study for the vector and scalar Hulthèn-type potentials was carried out by Dominguez-Adame and Chen et al. [24, 25]. The same potential for the s-wave KFG equation was obtained for regular and irregular boundary conditions in Ref. [26]. The path integral approach to the Green function for the KFG equation with these two potentials was studied in Ref. [27]. It should also be noted that the KFG equation has been recently studied with various combined potentials, for example, Hulthén plus ring-shaped potentials [28] and Manning-Rosen plus a Class of Yukawa potentials [29] by using ordinary and SUSY quantum mechanics.In Ref. [30–35], the scalar potential is equal and not equal to the vector potential which was assumed to get the bound states of the KFG equation with some typical potential by using the ordinary quantum mechanics. Furthermore, the KFG equation with the ring-shaped-like potential was investigated by Dong and Lozada-Cassou [34]. If we consider the case where the interaction potential is not enough to create particle-antiparticle pairs, the KFG equation can be applied to the treatment of a zero-spin particle and apply the Dirac equation to that of a 1/2-spin particle. When a particle is in a strong field, the relativistic wave equations should be considered in the quantum system. In any case, we can make the correction easily for nonrelativistic quantum mechanics.Since the Coulomb potential is one of the exactly solvable potentials in physics, it has a wide range of application to the several research areas such as nuclear and particle, atomic, condensed matter, and chemical physics. It is a natural and interesting question to study the relativistic effects for a particle within this potential especially for strong coupling. Notice that the KFG equation cannot be solved exactly for many potentials of interactions for nonzero angular momentuml≠0-states because of the centrifugal term of potentials.Noncentral potentials which play an important role for describing the quantum systems have the following form in spherical coordinates:(1)Vr,θ,ϕ=Vr+fθ,ϕr2,where Vr and fθ,ϕ are certain functions of their arguments. Coulomb or harmonic oscillator potentials are frequently used for the central part of equation (1).One of the convenient combinations is the Hartmann potential [15] for which
(2)V=ησ22a0r−ηa02r2sin2θε0,here, a0 is the Bohr radius, a0=ℏ2/me2, ε0 is the ground state energy of the hydrogen atom, ε0==me4/2ℏ2, and η and σ are positive real numbers. This potential was proposed by Hartmann in the framework of the nonrelativistic quantum mechanics for describing the organic molecules such as benzene within quantum chemistry. One can use the same potential in nuclear physics for studying the interactions between deformed nuclei.Other significant noncentral potentials of the form (1) were considered by Hautot [16] in the framework of the nonrelativistic quantum mechanics to study the problem of the motion of a charged particle. He considered the two- and three-dimensional harmonic oscillator potentials and the Coulomb potential to which terms of the type fθ/r2 were added and found such functions fθ that enabled him to solve exactly the corresponding Schrödinger equations. The original Hautot potential is defined in the following form:
(3)Vr,θ=μeVr+fθr2,here, μ is the mass of the charged particle and e is the charge of the particle.Hence, the Hartmann potential is a special case of the Hautot potentials.One needs to study the noncentral potentials rather than the central ones for getting better results in molecular structures and interactions. Examples to these situations include the use of the ring-shaped potentials in quantum chemistry to describe the ring-shaped organic molecules and in nuclear physics to investigate the interaction between the deformed nucleus and the spin-orbit coupling for a motion of the particle in the potential fields. This noncentral potential is also used as a mathematical model in the description of diatomic molecular vibrations, and it constitutes a convenient model for other physical situations.Coulomb plus ring-shaped potential was studied for the three-dimensional motion of a charged relativistic quantum particle in a noncentral potential [36]. This investigation is based on a finite-difference version of relativistic quantum mechanics (see references in [5]) and is a generalization of the results of [16] to the relativistic case. We emphasize that the Coulomb plus ring-shaped-like potentials (3) are widely used in various fields of physics: in nuclear and particle physics, in atomic and molecular physics, and in condensed matter and chemical physics in the nonrelativistic, as well as in the relativistic regions.Possible application of the combined potentials can be the study of the interaction between deformed pair of the nucleus and spin-orbit coupling in nuclear physics. Another possibility in the hadronic systems may be the description of vibrations which can shed light on other physical situations. In conclusion, the study of the wave functions and the energies in the bound and continuum states of the interacting systems with the linear combination of Coulomb and ring-shaped potentials is quite impressive, and it can provide a deeper and genuine appreciation of the physical properties.The remainder of the present work proceeds in the following order: in Section2, we give the information about the relativistic model of the Coulomb plus ring-shaped-like potential. Next, we present the solutions of the radial KFG equation in Section 3. Then, in Section 4, we give the dynamical group of symmetry of the present system. In Section 5, we present the solution of the angle-dependent part of the KFG equation. In Section 6, we obtain the nonrelativistic limit of the eigenfunctions and the energy spectrum, and some concluding remarks are stated in Section 7.


## 2. The Relativistic Model of the Coulomb plus Ring-Shaped-Like Potential

The KFG equation withSr scalar and Vr vector potentials has the form [1]
(4)−ℏ2c2∇2+Mc2+Sr2−E−Vr2ψr=0.In the nonrelativistic limit, this equation transforms into the Schrödinger equation for the sum of potentialsVN and SN, i.e.,
(5)−ℏ22MΔ+VNr+SNr−ENψNr=0,where EN=limc→∞E−Mc2, VNr=limc→∞Vr, and SNr=limc→∞Sr.We present here the exact solutions of the KFG equation with equal scalar and vector potentials for the sum Coulomb and ring-shaped potentials of the type (1). We also use an approach based on the Lie algebra of the group SU1,1, well known to be the dynamical group for several quantum systems [37–40]. We introduce a tilting transformation that relates between the physical states and the group sates which constitute a basis of the relevant unitary irreducible representation of SU1,1.As done in the literatures [30–35], here, we assume that
(6)Sr=Vr=−αEr+fθ,ϕr2.In terms of dimensionless variablesρ=r/ƛ, f0=Mfθ,ϕ/ℏ2 and parameters α0=αE/ℏc, ε=E/Mc2, equation (4) takes the form
(7)Hψ≡−∇ρ2+21+ε−α0ρ+f0ρ2+1−ε2ψρ,θ,ϕ=0.Here, the operator∇ρ2 has the usual form
(8)∇ρ2=∂ρ2+2ρ∂ρ−L^2ρ2,∂ρ=∂∂ρ,ρ=ρ,θ,ϕ,where L^2 is the square of the angular momentum operator, and it is defined as
(9)L^2=−Δθ,ϕ=−1sinθ∂θsinθ∂θ+1sin2θ∂ϕ2.Equation (7) does not differ in form from the Schrödinger equation (5). It allows separation of variables in the spherical coordinates ρ,θ,ϕ0≤ρ<∞,0≤θ≤π,0≤ϕ<2π.Searching the wave function in the form(10)ψρ,θ,ϕ=ρ−1RρFθ,ϕ,one obtains the set of separated differential equations
(11)∂ρ2+21+εα0ρ−gρ2−1+ε2R=0,(12)A^−gFθ,ϕ=0,here, g is a separation constant and
(13)A^=−Δθ,ϕ+M0fθ,ϕ,M0=2M1+εℏ2.The operatorA^ depends on energy, so its eigenvalues g will also depend on energy, g=gε. These dependences lead to the fact that an equation determining the energy levels (see formula (67)) will be very complicated. However, in the nonrelativistic limit, these dependences disappear, i.e.,
(14)limc→∞A^=A^N=−Δθ,ϕ+4MfNθ,ϕℏ2,limc→∞gε=gN.The operatorA^ commute with the Hamiltonian H, i.e., H,A^=0. It is responsible for separability of H in the spherical coordinates.


## 3. The Solutions of the Radial Klein-Fock-Gordon Equation

Let us go to the radial wave equation (11). Here, we suppose that the ε and g are arbitrary parameters. Putting
(15)Rρ=ρνe−1−ε2ρΩρ,ν=12+14+gin (11), there we receive the following equation for the function Ωρ:
(16)ρΩ″+2ν−1−ε2ρΩ′+21+εα0−21−ε2νΩ=0.If we compare this equation with the following equation(17)zu″+γ−zu′−αu=0,for a confluent hypergeometric function, its solutions are expressed in the form:
(18)Ωρ=Fν−1+εα01−ε2,2ν;21−ε2ρ.Consequently,(19)Rρ=ρνe−1−ε2ρFν−ε+1ε−1α0,2ν;21−ε2ρ.To analyze the obtained solutions, we consider separately the cases whenε<1 (or E<Mc2) and ε>1 (or E>Mc2).
(1)
Let beε<1. From the condition that the wave function is finite at zero R0=0 follows that ν>0. This case corresponds to the discrete energy spectrum of our system. Let us find the energy spectrum and wave function. Demand R∞=0 for the wave function (19) leads to the following condition for energy quantization:
(20)ν−ε+1ε−1α0=−n,n=0,1,2,⋯,where n denotes the radial quantum number.It follows from (20) the discrete energy level equation for our system in the case of 0<ε<1(21)n+ν1−ε=α01+ε,n=0,1,2,⋯,where the parameter ν may only take special values to be determined from (12). We emphasize that for hydrogen-like atoms, α0 must be replaced by Zα0. Then for the energies −1<ε<0, equation (21) will be satisfied for sufficiently large values Z. For example, when n=0,ν=1, we get Z=ν/α01+ε/1−ε. From here when ε=0.2, we find Z≥167.Thus, it follows from (19) and (20) that the radial wave functions corresponding to discrete energy levels (21) will have the form
(22)Rnρ=C~nρνe−1−εn2ρF−n,2ν;21−εn2ρ,here, εn are roots of equation (21). Using formulas [41]:
(23)Lnαx=α+1nn!F−n,α+1;x,the functions Rnρ can be expressed by the associated Laguerre polynomials. As a result, we conclude that the normalized radial wave functions, corresponding to the discrete energy spectrum (21), are
(24)Rnρ=Cnρνe−1−εn2ρLn2ν−121−εn2ρ,where Cn is the normalization constant
(25)Cn=2νn!1−εn2ν+1/22n+νΓn+2ν.It is found from the following condition(26)∫0∞Rn2ρdρ=1.To calculate this integral, the standard trick was used, i.e., recurrence relation(27)xLnαx=2n+α+1Lnαx−n+1Ln+1αx−n+1Ln−1αxand orthogonality properties for associated Laguerre polynomials:
(28)∫0∞xαe−xLnαxLmαdx=Γn+α+1n!δnm.(2)
Let nowε>1. In this case, the energy spectrum will be continuous, and the corresponding wave functions are obtained from expression (19). For example, for ε>1 (in this case ν is a real parameter)
(29)Rρ=Cρνe−iτρFν+i1+εα0ε2−1,2ν;2iε2−1⋅ρ,τ=ε2−1.Radial wave functions corresponding to energy valuesε=0,±1, can be obtained from (19) by using the corresponding passage to the limit. For example, for the value ε=1, we find
(30)limε→1Rρ=C′ρ−ν1+1J2ν1−14α0ρ,where ν1=limε→1ν, Jνz are well known Bessel functions. In the derivation of (30), we used the easily proved limit formula
(31)limα→0Fν−aα,γ;αx=Γγ+1ax−γ+1Jγ−12ax.The asymptotic behavior of Bessel functions at zero and at infinity is given by the formulas, respectively [41].
(32)Jνzz→0≈z2ν,Jνzz→∞≈2πzcosz−νπ2−π4−1zν2−14sinz−νπ4−π4.It follows that the wave function (31) is nonquadratic integrable at zero and at infinity. This means that the radial equation (11) does not have quadratically integrable solutions; therefore, a charged particle (e.g., electron) in the Coulomb plus ring-shaped potential field (6) of an arbitrary charge αE=Ze2 does not have bound states with energy E=Mc2, i.e., with zero binding energy. The nature of this phenomenon, as in the case of the Dirac equation considered in [42], is associated with the long-range nature of the Coulomb potential. The fact is that this potential generates an infinite set of bound states with energy equations defined by formula (21). This set accumulates at a point E=Mc2 but does not reach this point. Such a picture remains valid for any value of the charge Z. This conclusion can also be obtained based on the wave function (29).


## 4. Dynamical Symmetry Group

Let us now consider the radial equation (11) by the help of SU1,1 Lie algebra. The generators of SU1,1 algebra may be realized as [37]
(33)K0≡Γ0=12−ρ∂ρ2+gρ+ρ,K1≡Γ4=12−ρ∂ρ2+gρ−ρ,K2≡T=−iρ∂ρ.By a direct check, one can verify that these operators satisfy the commutation relations(34)Γ0,Γ4=iT,T,Γ0=iΓ4,Γ4,T=−iΓ0.The Casimir operator [38] is
(35)C2=Γ02−Γ42−T2=ss−1.We denote the states of a positive discrete series asn,s> such that
(36)Γ0n,s>=n+sn,s>,C2n,s>=ss−1n,s>,where s is the Bargmann index, s>0, and n=0,1,2,⋯. It should be noted that in our case from equation (33), we obtain C2=g=νν−1, so s=ν. Equation (11) can be written with the help of generators (33) as
(37)2−ε2Γ0+ε2Γ4−21+εα0R=0.Let us introduce a supplementary parameterθ and perform a tilting transformation
(38)R~=SR,S=e−iθT.From the commutation relations (34) and the formula
(39)eA^B^e−A^=B^+A^,B^+12!A^,A^,B^+⋯,where A^ and B^ are any two operators, it follows that
(40)e−iθTΓ0eiθT=Γ0coshθ+Γ4sinhθ,e−iβΓ0Γ4eiβΓ0=Γ4cosβ+Tsinβ,e−iθTΓ4eiθT=Γ4coshθ+Γ0sinhθ,e−iαΓ4Γ0eiαΓ4=Γ0coshα−Tsinhα,e−iβΓ0TeiβΓ0=Tcosβ−Γ4sinβ,e−iαΓ4TeiαΓ4=Tcoshα−Γ0sinhα.Through using formula (40), it is easy to verify that equation (37) becomes
(41)2−ε2coshθ+ε2sinhθΓ0+2−ε2sinhθ+ε2coshθΓ4−21+εα0R~=0.To solve equation (41) in an algebraic way, we consider separately the cases when ε<1 (or E<Mc2) and ε>1 (or E>Mc2).
(1)
Whenε<1 (discrete spectrum) in (41), a compact generator Γ0 can be diagonalized. Setting the coefficient of Γ4 to zero, we obtain tanhθ=−ε2/2−ε2 or eθ=1−ε2. As a result, we have
(42)1−ε2Γ0−1+εα0R~=0.It follows from (42) the energy level equation (21).
(2)
Whenε>1, noncompact generator Γ4 is diagonalized, having a continuous real spectrum λ∈R. In this case, we equate to zero the coefficient of the operator Γ0, i.e., 2−ε2coshθ+ε2sinhθ=0. From here, we get tanhθ=−2−ε2/ε2 and eθ=−1−ε2. Moreover, equation (41) takes the form
(43)−1−ε2Γ4−1+εα0R~1=0.Consequently,(44)Eλ=λ2+α02λ2−α02Mc2.We emphasize that the generatorsΓ0, Γ4, and T have no nonrelativistic limit: they diverge at c⟶∞. Generators Γ0′, Γ4′, and T′, which have the correct nonrelativistic limit, can be obtained from Γ0, Γ4, and T by the help of unitary transformation (compare with [43]). Unitary operator U, performing such a transformation, has the form
(45)U=e−iωT,tanhω=1−ε2−α021−ε2+α02,eω=1−ε2α0.Then, we will have(46)Γ0′=UΓ0U−1=Γ0coshω+Γ4sinhω=12−ξ∂ξ2+gξ+ξ,Γ4′=UΓ4U−1=Γ4coshω+Γ0sinhω=12−ξ∂ξ2+gξ−ξ,T′=UTU−1=T=−iξ∂ξ.In the formulas (46), the following dimensionless variable is introduced ξ=r/a0, where a0=ℏ2/MαE is a Bohr radius. It is associated with ρ=r/ƛ in the following way ξ=α0ρ. We can also write relations (46) in matrix form
(47)Γ0′Γ4′T′=coshωsinhω0sinhωcoshω0001Γ0Γ4T.As known, under unitary transformations, the Casimir operator remains unchanged, i.e.,C2′=UC2U−1=C2=g.Equation (11) can also be solved algebraically using the generators Γ0′, Γ4′, and T′. To do this, we rewrite it in the form
(48)α1Γ0′+α2Γ4′−21+εα0R=0,α1,2=α02±1−ε2α0.We now perform the tilting transformation to remove the noncompact generatorΓ4 [37–40]. To this end, we define
(49)R~′=S′R,S′=e−iθ′T.In view of formulas (40), we rewrite equation (48) in the form
(50)α1coshθ′+α2sinhθ′Γ0′+α1sinhθ′+α2coshθ′Γ4′−21+εα0R~′=0.If we choose nowθ′ as
(51)tanhθ′=−α2α1=1−ε2−α021−ε2+α02,eθ′=1−ε2α0,which corresponds to the case 0<ε<1; then, in (50), the generator Γ0′ is diagonalized, i.e.,
(52)1−ε2Γ0′−1+εα0R~′=0.We obtain from (52) again the discrete energy spectrum (21) pure algebraically.Now we give the relationship between the functionsR~ and R~′. It has the form
(53)R~′=S~R~,S~=SS′−1=e−iθ~T,θ~=θ−θ′,moreover, tanhθ~=α02−1/α02+1 and eθ~=α0. Consequently,
(54)R~′nξ=e−iθ~TR~nρ=R~nα0ρ.Whenε>1, we can diagonalized generator Γ4′, and we get the equation
(55)ε2−1Γ4′−1+εα0R~′=0.From here follows a continuous energy spectrum (44).


## 5. The Solutions of the Angular Equation

We now investigate the solutions of the angle-dependent equation (12). For this aim, we choose a function fθ,ϕ in the form
(56)fθ,ϕ=γ+βcosθ+αcos2θsin2θ.In this case,z-projection of the angular momentum operator L^z=−i∂ϕ will also commute with the Hamiltonian H. Thus, the variables in equation (12) can be further separated in the usual way
(57)Fθ,ϕ=Θθeimϕ2π.Here,m is the usual magnetic quantum number and is integer. In terms of the new variable x=cosθ, we obtain the following equation for the function Θθ≡Θx:
(58)∂x2+τ~xσx∂x+σ~xσ2xΘx=0,where σ=1−x2, τ~=−2x,σ~=−c2x2−c1x+c0,
(59)c2=g+M0α,c1=M0β,c0=g−m2−M0γ.The orthonormalized solutions of equation (58) were found in [36]. We give their explicit form
(60)Θkmθ=ckmsin2θ2−A1cos2θ2−A2Pk−2A1,−2A2cosθ,k=0,1,2,⋯.ParametersA1,2<0 are defined by expressions
(61)A1,2=−12m2+M0α+γ±β.Because theM0=2M1+ε/ℏ2, then these parameters, in contrary to the results of [36], depend on energy. A function Pkα,βx, which is the Jacobi polynomials, satisfies the following orthogonality condition [44].
(62)∫−111−xα1+xβPkα,βxPk′α,βxdx=dk2δkk′,where the square of the norm of the polynomials is
(63)dk2=2α+β+1Γk+α+1Γk+β+12k+α+β+1k!Γk+α+β+1,α>−1,β>−1.It follows from condition (62) that the angular wave functions (60) for distinct values of k satisfy orthonormal condition
(64)∫−11ΘkmxΘk′mxdx=δkk′.Here, the normalization constantckm is equal to
(65)ckm=k−A1−A2+1/2k!Γk−2A1−2A2+1Γk−2A1+1Γk−2A2+1.As shown in Ref. [36], separation constant g depends on k and m in this form
(66)g≡gkmε=kk+1+m2+M0α+γ+β+m2+M0α+γ−β+12m2+M0α+γ+β+m2+M0α+γ−β+12m2+M0α+γ+βm2+M0α+γ−β+12m2+M0γ−α.Thus, the exact discrete energy eigenvalues of the KFG equation for our system are defined as follows:(67)n+12+14+gkmε1−ε=α01+ε.This equation implies that the calculation of the energy levels becomes very complicated. We consider two special cases of formulas (21) and (60).
(1)
Forfθ,ϕ=0 in equation (6), the potential we used becomes Coulomb potential. In this case, g=ll+1, ν=l+1, and we will have A1=A2=−m/2, l=k+m,
(68)Θlmθ=−1m−m/212ml!2l+12l−m!l+m!sinθmPl−mm,mcosθ.Lettingα=β=γ=0 in (67) and solving the corresponding energy equation, we obtain for the Coulomb potential
(69)EnlMc2=εnl=1−2α02α02+ℏ2c2n+l+12.At the nonrelativistic limit, we get the following formula(70)limc→∞En−Mc2=ENn=−2MαE2ℏ2n+l+12.The corresponding radial wave functions have the form(71)Rnlρ=n!1−εn2l+3/2n+l+1Γn+2l+22ρl+1e−1−εn2ρLn2l+121−εn2ρ.Thanks to the connecting formula [41] between the Jacobi polynomials Pnα,αx and the Gegenbauer polynomials Cnλx(72)λ+12nCnλx=2λnPnλ−1/2,λ−1/2x,λn=Γn+λΓλ,we can express the functions Θlmθ (68) through the polynomials Cnλx, i.e.,
(73)Θlmθ=−1m−m/22mΓm+122l+12πl−m!l+m!sinθmCl−mm+1/2cosθ.On the other hand, there is the connecting formula [41]
(74)Plmcosθ=−2mπΓm+12sinθmCl−mm+1/2cosθbetween the Gegenbauer polynomials Cnλcosθ and the associated Legendre functions Plmcosθ. We have instead of the formula (73) [1]
(75)Θlmθ=−1m+m/22l+12l−m!l+m!Plmcosθ.Hence, the total wave functions take the form(76)ψnkmr,θ,ϕ=Cn21−εn2ρve−1−εn2ρLn2v−121−εn2ρsin2θ2−A1cos2θ2−A2Pk−2A1,−2A2cosθeimϕin the case of a discrete spectrum, and
(77)ψkmr,θ,ϕ=Cρve−iτρFv+2iε+1ε−1α0,2v;2iε2−1ρsin2θ2−A1cos2θ2−A2Pk−2A1,−2A2cosθeimϕin the case of a continuous spectrum.
(2)
Atγ=β=0, equation (60) coincides with equation (16) of Ref. [34]. (The relationship between our parameter ν and parameter L of Ref. [34] is as follows ν=L+1.)


## 6. The Nonrelativistic Limit

Let us now show that the energy spectrum and the radial wave functions considered in previous sections in the nonrelativistic limitc⟶∞ pass into the energy spectrum and the radial wave functions of the hydrogen-like atoms, respectively. We have
(78)limc→∞A1,2=A1,2ε=1=AN1,2,limc→∞ν=νε=1=νN,limc→∞21−εn2ρ=2ρNn+νN,ρN=2MαEℏ2r,limc→∞En−Mc2=ENn=−2MαE2ℏ2n+νN2,limc→∞Eλ−Mc2=ENλ=2MαE2ℏ2λ2,limc→∞ε+1ε−1α0=2MαE2ℏ2ENλ2≡1k,limc→∞Cn=CNn=n!2n+νNΓn+2νN,limc→∞2iε2−1ρ=2ikρN.Accounting equation (78), we obtain that the nonrelativistic limits of wave functions are equal to
(79)limc→∞Rnρ=RNnρN=CNn2ρNn+νNνNe−ρN/n+νNLn2νN−12ρNn+νN,limc→∞Rρ=RNρN=CNρNνNe−ikρNFνN+ik,2νN;2ikρN.We now give the nonrelativistic limit of operators (46)
(80)Γ0N=limc→∞Γ0′=12−ξ∂ξ2+gNξ+ξ,Γ4N=limc→∞Γ4′=12−ξ∂ξ2+gNξ−ξ,TN=limc→∞T′=T=−iξ∂ξ.


## 7. Conclusion

In this paper, the exact solutions of the KFG equation are obtained for equal scalar and vector Coulomb plus ring-shaped potentials. The KFG equation coincides in form with the Schrödinger equation for this potential. We solved the wave equation in the standard way by the separation of variables. For the radial wave equation, we constructed a dynamical symmetry groupSU1,1, which in turn allowed us to find the corresponding discrete and continuous energy spectrum of the system by purely algebraically. We also found an explicit form of the wave functions corresponding to the discrete and continuous energy spectra. They were expressed via associated Laguerre polynomials or by confluent hypergeometric functions.It is found that the radial wave functions, corresponding to both discrete and continuous energy eigenvalues, have the correct nonrelativistic limits. Angular wave functionsΘkmθ are expressed in terms of Jacobi polynomials, and they are a generalization of the result of [16] to the relativistic case. Therefore, in the nonrelativistic limit, wave functions Θkmθ reproduce the results in Ref. [16].It was also shown that the radial part of the equation of the motion possessesSU1,1 dynamical symmetry groups. In this line, our work may enable to provide a promising avenue in many branches of physics.In the literature, there are special cases of our results. For instance, (1) atfθ,ϕ=0 from our results, one obtains the known results of the corresponding problem (see for example [1]); (2) at β=γ=0 from formulas (21) and (24) for the discrete energy spectrum and radial wave functions, the corresponding formulas of Ref. [34] are obtained; and (3) at β=α=0, we obtain from our formulas (21) and (22) the corresponding formulas (11) and (15) of Ref. [45].The main results of this paper are the explicit and closed form expressions for the energy spectrum and corresponding wave functions. We have also shown the radial part of the equation of the motion which possessesSU1,1 dynamical symmetry groups.



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