Approximate Solutions to the Klein-Fock-Gordon Equation for the sum of Coulomb and Ring-Shaped like potentials

We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass$M$, described by the Klein-Fock-Gordon equation with equal scalar $S(\vec{r})$ and vector $V(\vec{r})$ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at $\left|E\right|Mc^{2} $ energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group $SU(1,1)$ 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\to \infty $ go over into the corresponding expressions for the nonrelativistic problem.


I. INTRODUCTION
Nonrelativistic Schrödinger and relativistic Dirac, Klein-Fock-Gordon (KFG), and finitedifference equations describe the systems in the nuclear physics, elementary physics as well as the in atomic and molecular physics. [1][2][3][4][5][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 include such as the Kratzer, [7] Morse, [8] Eckart, [9] Manning-Rosen, [10] Pöschl-Teller, [11] Hulthèn, [12] Wood-Saxon, [13] Makarov, [14] Hartmann, [15] and Hautot potentials [16]. 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 non-zero angular momentum. [17][18][19][20][21][22][23] The numerous interesting works devoted to the study of 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 use of the standard methods. Afterwards, the further study for the vector and scalar Hulthèn type potentials were carried out by Adame et al. [24,25] The same potential for s-wave KFG equation were 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].
In Ref. [28][29][30][31][32][33], the scalar potential is equal and not equal to the vector potential were assumed to get the bound states of the KFG equation with some typical potential by using the ordinary quantum mechanics. Furthermore, KFG equation with the ring-shaped like potential was investigated by Dong et al. [32] 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 non-relativistic quantum mechanics.
Since the Coulomb potential is one of the exactly solvable potentials in physics, it has wide range of application to the several research areas such as nuclear and particle, atomic, condensed matter, and chemical physics. It is natural and interesting question to study the relativistic effects for a particle within this potential especially for strong coupling. Notice that KFG equation cannot be solved exactly for many potentials of interactions for nonzero angular momentum l = 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 where V (r) and f (θ, ϕ) are certain functions of their arguments. Coulomb or harmonic oscillator potentials are frequently used for the central part of the equation (1.1). One of the convenient combinations is the Hartmann potential [15] for which here a 0 is the Bohr radius, a 0 = 2 /me 2 , ε 0 is the ground state energy of the hydrogen atom, ε 0 == me 4 /2 2 , 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 to describing organic molecules in quantum chemistry like benzene. One can use the same potential in nuclear physics for studying the interactions between deformed nuclei. Other significant noncentral potentials of the form (1.1) were considered by Hautot [16] in the framework of the nonrelativistic quantum mechanics to study the problem of the motion of a charged particle.
The original Hautot potential is defined in the following form: here µ is the mass of the charged particle, e is the charge of the particle.
He considered the two-and three-dimensional harmonic oscillator potentials and the Coulomb potential to which terms of the type were added and found such functions that enabled him to solve exactly the corresponding Schrdinger equations. 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 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 [34]. 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 (1.2) are widely used in various fields of physics: in nuclear and particle physics, in atomic and molecular physics, 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 be- The remainder of the present work proceeds in the following order: in Section II, 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 III. Then in Section IV, we give the dynamical group of symmetry of the present system. In Section V, we present the solution of the angle-dependent part of the KFG equation. In Section VI, we obtain the nonrelativistic limit of the eigenfunctions and the energy spectrum, and some concluding remarks are stated in Section VII.

LIKE POTENTIAL
The KFG equation with S(r) scalar and V (r) vector potentials has the form[1] In the nonrelativistic limit, this equation transforms into the Schrödinger equation for the sum of potentials V N and S N , i.e., 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 (1.1). We also use an approach based on the Lie algebra of the group SU(1, 1), well known to be the dynamical group for several quantum systems. [35][36][37][38] 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 SU(1, 1).
As done in the literatures [28][29][30][31][32][33], here we assume that Here, the operator ∇ 2 ρ has the usual form whereL 2 is the square of the angular momentum operator, and it is defined aŝ

Equation (2.4) does not differ in form from the Schrödinger equation (2.2). It allows sepa-
Searching the wave function in the form one obtains the set of separated differential equations here g is a separation constant and The operatorÂ depends on energy, so its eigenvalues g will also depend on energy, g = g(ε). These dependences lead to the fact that an equation determining the energy levels (see formula (5.12)) will be very complicated. However, in the nonrelativistic limit, these dependences disappear, i.e.
The operatorÂ commute with the Hamiltonian H, i.e. [H,Â] = 0. It is responsible for separability of H in the spherical coordinates.

III. THE SOLUTIONS OF THE RADIAL KLEIN-FOCK-GORDON EQUATION
Let us go to the radial wave equation (2.7). Here we suppose that the εand g are arbitrary parameters. Putting in (2.7), there we receive the following equation for the function Ω(ρ): If we comparing this equation with the following equation for a confluent hypergeometric function u = F (α; γ; z), its solutions are expressed in the form: Consequently, To analyze the obtained solutions, we consider separately the cases when |ε| < 1 (or |E| < Mc 2 ) and |ε| > 1 (or |E| > Mc 2 ).
1) Let be |ε| < 1. From the condition that the wave function is finite at zero R(0) = 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 (3.4) leads to the following condition for energy quantization where n denotes the radial quantum number.
It follows from (3.5) the discrete energy levels equation for our system in the case of where the parameter ν may only take special values to be determined from (2.8). We emphasize that for hydrogen-like atoms α 0 must be replaced by Zα 0 . Then for the energies −1 < ε < 0 equation (3.6) will be satisfied for sufficiently large values Z. For example, when From here when |ε| = 0.2 we find Z ≥ 167. Thus, it follows from (3.4) and (3.5) that the radial wave functions corresponding to discrete energy levels (3.6) will have the form here ε n are roots of equation (3.6). Using formulas [39]: the functions R n (ρ) 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 (3.6) are where C n is the normalization constant C n = 2 ν n ! (1 − ε 2 n ) ν+1/2 2(n + ν)Γ(n + 2ν) . (3.10) It is found from the following condition ∞ 0 R 2 n (ρ)dρ = 1. (3.11) To calculate this integral, the standard trick was used, i.e., recurrence relation and orthogonality properties for associated Laguerre polynomials: 2) Let now |ε| > 1. In this case, the energy spectrum will be continuous, and the corresponding wave functions are obtained from expression (3.4). For example, for ε > 1 (in this case ν is a real parameter) Radial wave functions corresponding to energy values ε = 0, ±1, can be obtained from (3.4) by using the corresponding passage to the limit. For example, for the value ε = 1 we find where ν 1 = lim ε→1 ν, a J ν (z) are well known Bessel functions. In the derivation of (3.15), we used the easily proved limit formula The asymptotic behavior of Bessel functions at zero and at infinity are given by the formulas, ) . This conclusion can also be obtained based on the wave function (3.14).

IV. DYNAMICAL SYMMETRY GROUP
Let us now consider the radial equation (2.7) by help of SU(1, 1) Lie algebra. The generators of SU(1, 1) algebra may be realized as [35] K The Casimir operator [36] is We denote the states of a positive discrete series as |n, s > such that Γ 0 |n, s >= (n + s) |n, s > , where s is the Bargmann index, s > 0 and n = 0, 1, 2, .... It should be note that in our case from Eq. (4.1), we obtain C 2 = g = ν (ν − 1), so s = ν. The equation (2.7) can be written with the help of generators (3.1) as whereÂ andB are any two operators, it follows that (4.8) Through using the formula (4.8), it is easily to verified that the equation (4.5) becomes To solve equation (4.9) in an algebraic way, we consider separately the cases when |ε| < 1 (or |E| < Mc 2 ) and |ε| > 1 (or |E| > Mc 2 ).
2) When |ε| > 1 non-compact generator Γ 4 is diagonalized, having a continuous real spectrum λ ∈ R. In this case, we equate to zero the coefficient of the operator Γ 0 , i.e.
Moreover, equation (4.9) takes the form Consequently, We emphasize that the generators Γ 0 , Γ 4 and T have no nonrelativistic limit: they diverge at c → ∞. Generators Γ ′ 0 , Γ ′ 4 and T ′ , which have the correct nonrelativistic limit can be obtained from Γ 0 , Γ 4 and T by help of unitary transformation (compare with [41]). Unitary operator U, performing such a transformation has the form Then, we will have 14) In the formulas (4.14) the following dimensionless variable is introduced ξ = r/a 0 , where a 0 = 2 /Mα e is a Bohr radius. It is associated with ρ = r/λ in the following way ξ = α 0 ρ.
We can also write relations (4.14) in matrix form As is known, under unitary transformations, the Casimir operator remains unchanged, i.e.
can also be solved algebraically using the generators Γ ′ 0 , Γ ′ 4 and T ′ . To do this, we rewrite it in the form We now perform the tilting transformation to remove the noncompact generator Γ 4 [35][36][37][38].
To this end, we defineR In view of formulas (4.8), we rewrite equation (4.16) in the form If we choose now θ ′ as which corresponds to the case 0 < |ε| < 1, then in (4.18) the generator Γ ′ 0 is diagonalized, i.e.
We obtain from (4.20) again the discrete energy spectrum (3.6) pure algebraically.

V. THE SOLUTIONS OF THE ANGULAR EQUATION
We now investigate the solutions of the angle-dependent equation (2.8). For this aim we choose a function f (θ, ϕ) in the form f (θ, ϕ) = γ + β cos θ + α cos 2 θ sin 2 θ .
In this case z-projection of the angular momentum operatorL z = −i∂ ϕ will also commute with the Hamiltonian H. Thus, the variables in equation (2.8) can be further separated in the usual way Here m is the usual magnetic quantum number and is integer. In terms of the new variable x = cos θ, we obtain the following equation for the function Θ(θ) ≡ Θ(x): where σ = 1 − x 2 ,τ = −2x,σ = −c 2 x 2 − c 1 x + c 0 , The orthonormalized solutions of equation (5.3) were found in [34]. We give their explicit (cos θ), k = 0, 1, 2, ...

(5.5)
Parameters A 1,2 < 0 are defined by expressions Because the M 0 = 2M(1 + ε)/ 2 , then these parameters, in contrary to the results of [34], are depend on energy. A function P (α,β) k (x), which are the Jacobi polynomials, satisfies the following orthogonality condition. [42] 1 −1 where the square of the norm of the polynomials is It follows from condition (5.7) that the angular wave functions (5.5) for distinct values of k satisfy orthonormal condition Here the normalization constant c km is equal to As shown in the Ref. [34], separation constant g depends on k and m in this form (5.11) Thus, the exact discrete energy eigenvalues of the KFG equation for our system are defined as follows (n + 1 2 This equation implies that the calculation of the energy levels becomes very complicated. We consider two special cases of formulas (3.6) and (5.5).
In this case g = l(l + 1), ν = l + 1 and we will have A 1 = A 2 = − |m| /2, l = k + |m|, Letting α = β = γ = 0 in (5.12) and solving the corresponding energy equation, we obtain for the Coulomb potential (5.14) At the nonrelativistic limit, we get the following formula The corresponding radial wave functions have the form Thanks to the connecting formula [39] between the Jacobi polynomials P (α,α) n (x) and the Gegenbauer polynomials C λ n (x) between the Gegenbauer polynomials C λ n (cos θ) and the associated Legendre functions P |m| l (cos θ). We have instead of the formula (5.18) [1] Θ lm (θ) = (−1) in the case of a discrete spectrum, and (5.22) in the case of a continuous spectrum.

VI. THE NONRELATIVISTIC LIMIT
Let us now show that the energy spectrum and the radial wave functions considered in previous sections in the nonrelativistic limit c → ∞ pass into the energy spectrum and the radial wave functions of the hydrogen like atoms respectively. We have Accounting the Eq. (6.1), we obtain that nonrelativistic limits of wave functions are equal We now give the nonrelativistic limit of operators (4.14)  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 functions Θ km (θ) 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 Θ km (θ) reproduce the results in Ref. [16].
It was also shown that radial part of the equation of the motion possesses SU(1, 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) at f (θ, 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 (3.6) and (3.9) for the discrete energy spectrum and radial wave functions, the corresponding formulas of Ref. [32] are obtained. 3) at β = α = 0 we obtain from our formulas (3.6) and (3.7) the corresponding formulas (11) and (15) of Ref. [43].
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 possesses SU(1, 1) dynamical symmetry groups.