Quasi-Analytical Solutions of DKP Equation under the Deng-Fan Interaction

Abstract

Quasianalytical solutions of Duffin-Kemmer-Petiau equation under scalar and vector Deng-Fan potentials are reported via a novel ansatz.

1. Introduction

A yet open challenge in the annals of wave mechanics is the equivalence or nonequivalence of Duffin-Kemmer-Petiau (DKP) and Klein-Gordon (KG) equations. While the latter solely describes relativistic spin-zero bosons, the former can investigate spin-one particles as well. This equation was introduced 1930s in search for a linear relativistic wave equation similar to Dirac equation for relativistic bosons [1–4]. In its present format, the DKP equation is normally represented in two five and ten-dimensional versions that respectively work for spin-zero and spin-one bosons. At the present, to our best knowledge, the equivalence of KG and spin-zero DKP equations is doubted [5–14]. This is also true for the other counterpart of DKP equation, that is the Proca equation which is a relativistic framework to study spin-one bosons. There are papers that investigate related problems via these equations and report motivating data [15–20]. Here, our focus is on the spin-zero version of the equation, which has many applications [21–25].

In solving wave equations of mechanics, various analytical techniques including supersymmetry quantum mechanics (SUSYQM), Nikiforov-Uvarov (NU), quantization rule, Lie algebras, WKB approximation, point canonical transformation (PCT), and series expansion have been applied [26–40]. Nevertheless, we face situations in which these methodologies do not work very well. A very successful approach in such cases is proposing a physical ansatz solution that is on the one hand consistent with the requirements of quantum mechanics and on the other hand satisfies the equation. This is in many cases a definitely cumbersome, and in some cases an impossible task. Here we follow this rich quasianalytical technique to solve the DKP equation under the Deng-Fan potential [41]. There are, however, a few papers which discuss this attractive potential. This is partially due to the complex structure of this interaction. As far as we know, the published works include references [42–46] that consider the potential under either relativistic or nonrelativistic equations. The structure of the present paper is as follows. In Section 2, we briefly introduce the DKP equation and a Pekeris-type approximation. In the next part, we introduce some transformations as well as an attractive ansatz by which the problem is solved in a quasianalytical manner.

2. The DKP Equation

The DKP Hamiltonian for scalar 𝑈_𝑠  and vector 𝑈^𝑜_𝑣  interactions is 𝛽.^→𝑝𝑐+𝑚𝑐²+𝑈_𝑠+𝛽⁰𝑈^𝑜_𝑣𝜓^→𝑟=𝛽⁰𝐸𝜓^→𝑟,(2.1) where 𝜓^→𝑟=𝜓_upper𝑖𝜓_lower,(2.2) and the upper and lower components, respectively, are 𝜓_upper≡𝜙𝜑,𝜓_lower≡𝐴₁𝐴₂𝐴₃.(2.3)

The engaged matrices are𝛽⁰=𝜃00_𝑇0,𝛽^𝑖=0𝜌^𝑖−𝜌^𝑖_𝑇0,(2.4a) with 0, 0 and 0, respectively, being 2×2,2×3,3×3  zero matrices 𝜃=0110,𝜌¹=−100000,𝜌²=0−10000,𝜌³=00−1000.(2.4b)

The equation, in (3+0)-dimensions, is written as [1–4] 𝑚𝑐²+𝑈_𝑠𝜙=𝐸−𝑈^𝑜_𝑣𝜑+ℏ𝑐^→∇.𝐴^→,_→∇𝜙=𝑚𝑐²+𝑈𝐴_𝑠^→,𝑚𝑐²+𝑈_𝑠𝜑=𝐸−𝑈^𝑜_𝑣𝜙,(2.5) where  𝐴^→=(𝐴₁,𝐴₂,𝐴₃). In (2.4a) and (2.4b), 𝜓 is a simultaneous eigenfunction of 𝐽²  and  𝐽₃, that is, 𝐽²𝜓_upper𝜓_lower=𝐿²𝜓_upper(𝐿+𝑆)²𝜓_lower()𝜓=𝐽𝐽+1^upper𝜓_lower,𝐽₃𝜓_upper𝜓_lower=𝐿₃𝜓_upper𝐿₃+𝑠₃𝜓_lower𝜓=𝑀^upper𝜓_lower,(2.6) and the general solution is considered as 𝜓_𝐽𝑀(𝑟)=𝑓_𝑛𝐽(𝑟)𝑌_𝐽𝑀(Ω)𝑔_𝑛𝐽(𝑟)𝑌_𝐽𝑀(Ω)𝑖∑_𝐿ℎ_𝑛𝐽𝐿(𝑟)𝑌^𝑀_𝐽𝐿1(Ω),(2.7) where spherical harmonics 𝑌_𝐽𝑀(Ω) are of order 𝐽, 𝑌^𝑀_𝐽𝐿1(Ω) are the normalized vector spherical harmonics, and 𝑓_𝑛𝐽,𝑔_𝑛𝐽,  and ℎ_𝑛𝐽𝐿  represent the radial wavefunctions. The above equations yield the following coupled differential equations [30–38]: 𝐸_𝑛,𝐽−𝑈⁰_𝑣𝐹_𝑛,𝐽(𝑟)=𝑚𝑐²+𝑈_𝑠𝐺_𝑛,𝐽(𝑟),𝑑𝐹_𝑛,𝐽(𝑟)−𝑑𝑟𝐽+1𝑟𝐹_𝑛,𝐽(𝑟)1=−𝛼_𝐽𝑚𝑐²+𝑈_𝑠𝐻_1,𝑛,𝐽(𝑟),𝑑𝐹_𝑛,𝐽(𝑟)+𝐽𝑑𝑟𝑟𝐹_𝑛,𝐽(𝑟)=1𝜁_𝐽𝑚𝑐²+𝑈_𝑠𝐻_{−1,𝑛,𝐽}(𝑟),−𝛼_𝐽𝑑𝐻_1,𝑛,𝐽(𝑟)+𝑑𝑟𝐽+1𝑟𝐻_1,𝑛,𝐽(𝑟)+𝜁𝑑𝐻_{−1,𝑛,𝐽}(𝑟)−𝐽𝑑𝑟𝑟𝐻_{−1,𝑛,𝐽}(𝑟)=1ℏ𝑐𝑚𝑐²+𝑈_𝑠𝐹_𝑛,𝐽(𝑟)−𝐸_𝑛,𝐽−𝑈⁰_𝑣𝐺_𝑛,𝐽(𝑟),(2.8) which give [39–41] 𝑑²𝐹_𝑛,𝐽(𝑟)𝑑𝑟²𝜁1+^2𝐽𝛼²_𝐽−𝑑𝐹_𝑛,𝐽(𝑟)𝑈𝑑𝑟𝑚+^′𝑠𝑈_𝑠𝜁1+^2𝐽𝛼²_𝐽+𝐹_𝑛,𝐽(𝑟−𝐽())𝐽+1𝑟²𝜁1+^2𝐽𝛼²_𝐽+𝑈^′_𝑠𝑚+𝑈_𝑠𝐽+1𝑟−𝜁²_𝐽𝛼²_𝐽𝐽𝑟−1𝛼²_𝐽𝑚+𝑈_𝑠²−𝐸_𝑛,𝐽−𝑈⁰_𝑣²=0,(2.9) Where 𝛼_𝐽=(𝐽+1)/(2𝐽+1),𝑓_𝑛𝐽(𝑟)=𝐹(𝑟)/𝑟,𝑔_𝑛𝐽(𝑟)=𝐺(𝑟)/𝑟,ℎ_{𝑛𝐽𝐽±1}=𝐻_±1/𝑟  and  𝜁_𝐽=𝐽/(2𝐽+1). When 𝑈_𝑠=0, we recover the well-known formula [30–32] 𝑑²𝑑𝑟²−𝐽(𝐽+1)𝑟²+𝐸_𝑛,𝐽−𝑈⁰_𝑣²−𝑚²𝐹_𝑛,𝐽(𝑟)=0.(2.10)

We consider the Deng-Fan vector and scalar potentials [42–46] 𝑈_𝑠=𝑉₀+𝑉₁𝑒^𝛼𝑟+𝑉−1(𝑒^𝛼𝑟−²1)²,𝑈⁰_𝑣=𝑢₀+𝑢₁𝑒^𝛼𝑟+𝑢−1(𝑒^𝛼𝑟−²1)².(2.11) By a change of variable of the form 𝐹_𝑛,𝐽(𝑟)=𝑚+𝑈_𝑠𝑢_𝑛,𝐽(𝑟),(2.12)

(2.9) is written as 𝑑²𝑑𝑟²+𝑈_𝑠^′′21𝑚+𝑈_𝑠−34𝑈^′2_𝑠𝑚+𝑈_𝑠²−𝐽(𝐽+1)𝑟²+𝐶𝑈^′_𝑠𝐴𝑚+𝑈_𝑠1𝑟−1𝐴𝛼²_𝐽𝑚+𝑈_𝑠²𝐸_𝑛,𝐽−𝑈⁰_𝑣²𝑢_𝑛,𝐽(𝑟)=0,(2.13) where 𝜁𝐴=1+^2𝐽𝛼²_𝐽𝜁,𝐶=𝐽+1−^2𝐽𝛼²_𝐽𝐽.(2.14)

From (2.11), (2.13) can be written as 𝑑²𝑑𝑟²+12−𝛼²𝑉₁𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)²+2𝛼²𝑉₁𝑒^2𝛼𝑟(𝑒^𝛼𝑟−1)³−2𝛼²𝑉₂𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)³+6𝛼²𝑉₂𝑒^2𝛼𝑟(𝑒^𝛼𝑟−1)⁴×1(1/𝑚)+𝑉₀+𝑉₁/(𝑒^𝛼𝑟−1)+𝑉₂/(𝑒^𝛼𝑟−1)²−341𝑚+𝑉₀+𝑉₁/(𝑒^𝛼𝑟−1)+𝑉₂/(𝑒^𝛼𝑟−1)²²−𝛼𝑉₁𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)²−2𝛼𝑉₂𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)³²−𝐽(𝐽+1)𝑟²+𝐶𝐴𝑚+𝑉₀+𝑉₁/(𝑒^𝛼𝑟−1)+𝑉₂/(𝑒^𝛼𝑟−1)²−𝛼𝑉₁𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)²−2𝛼𝑉₂𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)³1𝑟−1𝐴𝛼²_𝐽𝑚+𝑉₀+𝑉₁𝑒^𝛼𝑟+𝑉−1(𝑒^𝛼𝑟−²1)²²−𝐸_𝑛,𝐽−𝑢₀−𝑢₁𝑒^𝛼𝑟−𝑢−1(𝑒^𝛼𝑟−²1)²²𝑢_𝑛,𝐽(𝑟)=0.(2.15)

Here, we use the following approximations for the centrifugal term [47]:1𝑟²≈𝛼²(𝑒^𝛼𝑟−1)²,1(2.16a)𝑟≈𝛼𝑒^𝛼𝑟.−1(2.16b)

Equation (2.16a) is a quite logical alternative for 𝛼<0.1  (see Figure 1), and (2.16a) and (2.16b) brings (2.15) into the form 𝑑²𝑑𝑟²+−𝛼²𝑉₁2𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)²+𝛼²𝑉₁𝑒^2𝛼𝑟(𝑒^𝛼𝑟−1)³−𝛼²𝑉₂𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)³+3𝛼²𝑉₂𝑒^2𝛼𝑟(𝑒^𝛼𝑟−1)⁴×1𝑚+𝑉₀+𝑉₁/(𝑒^𝛼𝑟−1)+𝑉₂/(𝑒^𝛼𝑟−1)²+−34𝛼²𝑉²₁𝑒^2𝛼𝑟(𝑒^𝛼𝑟−1)⁴−3𝛼²𝑉²₂𝑒^2𝛼𝑟(𝑒^𝛼𝑟−1)⁶−3𝛼²𝑉₁𝑉₂𝑒^2𝛼𝑟(𝑒^𝛼𝑟−1)⁵×1𝑚+𝑉₀+𝑉₁/(𝑒^𝛼𝑟−1)+𝑉₂/(𝑒^𝛼𝑟−1)²²−𝐽(𝐽+1)𝛼²1𝑒^𝛼𝑟−1²+𝐶𝛼𝐴1𝑒^𝛼𝑟−1−𝛼𝑉₁𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)²−2𝛼𝑉₂𝑒^𝛼𝑟(𝑒^𝛼𝑟−1)³×1𝑚+𝑉₀+𝑉₁/(𝑒^𝛼𝑟−1)+𝑉₂/(𝑒^𝛼𝑟−1)²−1𝐴𝛼²_𝐽𝑚+𝑉₀+𝑉₁𝑒^𝛼𝑟+𝑉−1(𝑒^𝛼𝑟−²1)²²−𝐸_𝑛,𝐽−𝑢₀−𝑢₁𝑒^𝛼𝑟−𝑢−1(𝑒^𝛼𝑟−²1)²²𝑢_𝑛,𝐽(𝑟)=0.(2.17)

Figure 1 

1/𝑟2and its approximation for different value of 𝛼.

By introducing  𝑧=𝑒^𝛼𝑟 and making the transformation  𝑢_𝑛,𝐽(𝑧)=𝜙_𝑛,𝐽(𝑧)/(√𝑧), we obtain 𝑑²𝑑𝑧²+141𝑧²+−𝑉₁2𝑚+𝑉₀1𝑧+𝑉₁𝑚+𝑉₀1−𝑉𝑧−1𝑚+²𝑉₀1𝑧()+𝑧−13𝑉₂𝑚+𝑉₀1(𝑧−1)²×1𝑧+𝑧₁𝑧+𝑧₂+−3𝑉²₁4𝑚+𝑉₀²−3𝑉²₂𝑚+𝑉₀²1(𝑧−1)²−3𝑉₂𝑉₁𝑚+𝑉₀²11𝑧−1(𝑧+𝑧₁)(𝑧+𝑧₂)²()1−𝐽𝐽+1𝑧²(𝑧−1)²+1𝑧+𝑧₁𝑧+𝑧₂−𝐶𝑉₁𝐴𝑚+𝑉₀1𝑧()−𝑧−12𝐶𝑉₂𝐴𝑚+𝑉₀1𝑧(𝑧−1)²−1𝐴𝛼²𝛼²_𝐽𝑚+𝑉₀²𝐸_𝑛,𝐽−𝑢₀²1𝑧²−1𝐴𝛼²𝛼²_𝐽𝑉²₁−𝑢²₁+2𝑉₂𝑚+𝑉₀+2𝑢₂𝐸_𝑛,𝐽−𝑢₀×1𝑧²(𝑧−1)²−1𝐴𝛼²𝛼²_𝐽2𝑉₁𝑚+𝑉₀+2𝑢₁𝐸_𝑛,𝐽−𝑢₀1𝑧²()−1𝑧−1𝐴𝛼²𝛼²_𝐽2𝑉₂𝑉₁−2𝑢₁𝑢₂1𝑧²(𝑧−1)³−1𝐴𝛼²𝛼²_𝐽𝑉²₂−𝑢²₂1𝑧²(𝑧−1)⁴𝜙_𝑛,𝐽(𝑧)=0, which after decomposition of fractions gives 𝑑²𝑑𝑧²+𝜆𝑧+𝜔𝑧²+𝑠+𝑞𝑧−1(𝑧−1)²+𝑝(𝑧−1)³+𝑁(𝑧−1)⁴+𝑔𝑧+𝑧₁+ℎ𝑧+𝑧₁²+𝜒𝑧+𝑧₂+𝜅(𝑧−1)²𝜙_𝑛,𝐽(𝑧)=0(2.19) with 𝑧₁=12𝑉₁−2𝑚+𝑉₀𝑚+𝑉₀+𝑉₁−2𝑚+𝑉₀𝑚+𝑉₀²−4𝑚+𝑉₀−𝑉₁+𝑉₂𝑚+𝑉₀,𝑧₂=12𝑉₁−2𝑚+𝑉₀𝑚+𝑉₀−𝑉₁−2𝑚+𝑉₀𝑚+𝑉₀²−4𝑚+𝑉₀−𝑉₁+𝑉₂𝑚+𝑉₀,𝑉𝜆=−¹2𝑧₁𝑧₂𝑚+𝑉₀+1𝑧₁𝑧₂𝑉₂𝑚+𝑉₀+𝐶𝑉₁𝐴𝑚+𝑉₀+2𝑉₁𝑚+𝑉₀𝐴𝛼²_𝐽𝛼²+2𝑢₁𝐸_𝑛,𝐽−𝑢₀𝐴𝛼²_𝐽𝛼²()−2−2𝐽𝐽+1𝐴𝛼²_𝐽𝛼²𝑉²₁−𝑢²₁+2𝑉₂𝑚+𝑉₀+2𝑢₂𝐸_𝑛,𝐽−𝑢₀+3𝐴𝛼²_𝐽𝛼²2𝑉₁𝑉₂−2𝑢₁𝑢₂−4𝐴𝛼²_𝐽𝛼²𝑉²₂−𝑢²₂−2𝐶𝑉₂𝐴𝑧₁𝑧₂𝑚+𝑉₀,1𝜔=4−1𝐴𝛼²_𝐽𝛼²𝑚+𝑉₀²−𝐸_𝑛,𝐽−𝑢₀²+2𝑉₁𝑚+𝑉₀𝐴𝛼²_𝐽𝛼²+2𝑢₁𝐸_𝑛,𝐽−𝑢₀𝐴𝛼²_𝐽𝛼²()−1−𝐽𝐽+1𝐴𝛼²_𝐽𝛼²𝑉²₁−𝑢²₁+2𝑉₂𝑚+𝑉₀+2𝑢₂𝐸_𝑛,𝐽−𝑢₀+1𝐴𝛼²_𝐽𝛼²2𝑉₁𝑉₂−2𝑢₁𝑢₂−1𝐴𝛼²_𝐽𝛼²𝑉²₂−𝑢²₂,𝑉𝑆=¹𝑚+𝑉₀𝑧₁+1𝑧₂+1+1𝑧₁+1𝑧₂−𝑉+1𝑚+²𝑉₀−𝐶𝑉₁𝐴𝑚+𝑉₀−3𝑉₂𝑧₁+𝑧₂+2𝑧₁+1²𝑧₂+1²𝑚+𝑉₀+6𝑉²₂𝑧₁+𝑧₂+2𝑧₁+1³𝑧₂+1³𝑚+𝑉₀²−3𝑉₁𝑉₂𝑚+𝑉₀²𝑧₁+1²𝑧₂+1²−2𝑉₁𝑚+𝑉₀𝐴𝛼²_𝐽𝛼²−2𝑢₁𝐸_𝑛,𝐽−𝑢₀𝐴𝛼²_𝐽𝛼²()+2+2𝐽𝐽+1𝐴𝛼²_𝐽𝛼²𝑉²₁−𝑢²₁+2𝑉₂𝑚+𝑉₀+2𝑢₂𝐸_𝑛,𝐽−𝑢₀−3𝐴𝛼²_𝐽𝛼²2𝑉₁𝑉₂−2𝑢₁𝑢₂+4𝐴𝛼²_𝐽𝛼²𝑉²₂−𝑢²₂+2𝐶𝑉₂𝑧₁𝑧₂+2𝑧₁+2𝑧₂+3𝐴𝑚+𝑉₀𝑧₁+1²𝑧₂+1²,𝑞=3𝑉₂𝑚+𝑉₀𝑧₁+1𝑧₂−+13𝑉²₂𝑚+𝑉₀²𝑧₁+1²𝑧₂+1²()−1−𝐽𝐽+1𝐴𝛼²_𝐽𝛼²𝑉²₁−𝑢²₁+2𝑉₂𝑚+𝑉₀+2𝑢₂𝐸_𝑛,𝐽−𝑢₀+2𝐴𝛼²_𝐽𝛼²2𝑉₁𝑉₂−2𝑢₁𝑢₂−3𝐴𝛼²_𝐽𝛼²𝑉²₂−𝑢²₂−2𝐶𝑉₂𝐴𝑚+𝑉₀𝑧₁+1𝑧₂,𝑉+1𝑔=−¹2𝑧₁𝑧₁−𝑧₂𝑚+𝑉₀+𝑉₁𝑧₁+1𝑧₁−𝑧₂𝑚+𝑉₀+1𝑧₁𝑧₁−𝑧₂𝑧₁𝑉+1𝑚+²𝑉₀+𝐶𝑉₁𝐴𝑚+𝑉₀−3𝑉₂𝑚+𝑉₀𝑧₁+1²𝑧₁−𝑧₂−6𝑉²₁4𝑚+𝑉₀²𝑧₁−𝑧₂³−6𝑉²₂−𝑧₂+2𝑧₁+1𝑚+𝑉₀²𝑧₁−𝑧₂³𝑧₁+1³+3𝑉₂𝑉₁−𝑧₂+3𝑧₁+2𝑚+𝑉₀²𝑧₁−𝑧₂³𝑧₁+1²−2𝐶𝑉₂𝐴𝑧₁𝑚+𝑉₀𝑧₁−𝑧₂𝑧₁+1²,1𝑝=−𝐴𝛼²_𝐽𝛼²2𝑉₁𝑉₂−2𝑢₁𝑢₂+2𝐴𝛼²_𝐽𝛼²𝑉²₂−𝑢²₂,1𝑁=−𝐴𝛼²_𝐽𝛼²𝑉²₂−𝑢²₂,ℎ=−3𝑉²₁4𝑚+𝑉₀²𝑧₂−𝑧₁²−3𝑉²₂𝑚+𝑉₀²𝑧₁−𝑧₂²𝑧₁+1²+3𝑉₁𝑉₂𝑚+𝑉₀²𝑧₁−𝑧₂²𝑧₁,𝑉+1𝜒=¹2𝑧₂𝑧₁−𝑧₂𝑚+𝑉₀+𝑉₁𝑧₂+1𝑧₂−𝑧₁𝑚+𝑉₀+1𝑧₂𝑧₂−𝑧₁𝑧₂𝑉+1𝑚+²𝑉₀+𝐶𝑉₁𝐴𝑚+𝑉₀−3𝑉₂𝑚+𝑉₀𝑧₂+1²𝑧₂−𝑧₁+6𝑉²₁4𝑚+𝑉₀²𝑧₁−𝑧₂³−6𝑉²₂−𝑧₁+2𝑧₂+1𝑚+𝑉₀²𝑧₂−𝑧₁³𝑧₂+1³+3𝑉₂𝑉₁−𝑧₁+3𝑧₂+2𝑚+𝑉₀²𝑧₂−𝑧₁³𝑧₂+1²−2𝐶𝑉₂𝐴𝑧₂𝑚+𝑉₀𝑧₂−𝑧₁𝑧₂+1²,𝜅=−3𝑉²₁4𝑚+𝑉₀²𝑧₂−𝑧₁²−3𝑉²₂𝑚+𝑉₀²𝑧₁−𝑧₂²𝑧₂+1²+3𝑉₁𝑉₂𝑚+𝑉₀²𝑧₁−𝑧₂²𝑧₂.+1

Let us now consider an ansatz of the form 𝜙_𝑛,𝐽(𝑧)=𝑓_𝑛(𝑧)exp𝑔_𝐽(𝑧),(2.21) with 𝑓_𝑛(𝑧)=1,𝑛=0,_𝑛∏_𝑖=1𝑧−𝛼^𝑛_𝑖𝑔,𝑛>0,_𝐽(𝑧)(𝑧)()+𝜂=𝛾ln+𝛽ln𝑧−1𝑧−1+𝜉ln𝑧+𝑧₁+𝛿ln𝑧+𝑧₂.(2.22) Substitution of the latter in (2.20) yields 𝜙^′′_0,𝐽(𝑧)=−2𝛽𝛾−2𝛾𝜂+2𝛾𝜉𝑧₁+2𝛾𝛿𝑧₂1𝑧+𝛾²1−𝛾𝑧²+2𝛽𝛾+2𝛾𝜂+2𝛽𝜉1+𝑧₁+2𝛽𝛿1+𝑧₂+2𝜂𝜉1+𝑧₁²+2𝜂𝛿1+𝑧₂²1+𝛽𝑧−1₂−𝛽−2𝛾𝜂−2𝜂𝜉1+𝑧₁−2𝜂𝛿1+𝑧₂1(𝑧−1)²+12𝜂−2𝛽𝜂(𝑧−1)³+𝜂²1(𝑧−1)⁴+−2𝛾𝜉𝑧₁−2𝛽𝜉1+𝑧₁−2𝜂𝜉1+𝑧₁²+2𝜉𝛿𝑧₂−𝑧₁1𝑧+𝑧₁+𝜉²1−𝜉𝑧+𝑧₁²+−2𝛿𝛾𝑧₂−2𝛽𝛿1+𝑧₂−2𝜂𝛿1+𝑧₂²−2𝜉𝛿𝑧₂−𝑧₁1𝑧+𝑧₂+𝛿²1−𝛿𝑧+𝑧₂²𝜙_0,𝐽(𝑧).

For the sake of simplicity, here we consider only the nodless solutions. Substitution of the proposed ansatz solution and equating the corresponding powers on both sides give −2𝛽𝛾−2𝛾𝜂+2𝛾𝜉𝑧₁+2𝛾𝛿𝑧₂𝛾=−𝜆,₂−𝛾=−𝜔,2𝛽𝛾+2𝛾𝜂+2𝛽𝜉1+𝑧₁+2𝛽𝛿1+𝑧₂+2𝜂𝜉1+𝑧₁²+2𝜂𝛿1+𝑧₂²𝛽=−𝑆,₂−𝛽−2𝛾𝜂−2𝜂𝜉1+𝑧₁−2𝜂𝛿1+𝑧₂𝜂=−𝑞,2𝜂−2𝛽𝜂=−𝑝,₂−=−𝑁2𝛾𝜉𝑧₁−2𝛽𝜉1+𝑧₁−2𝜂𝜉1+𝑧₁²+2𝜉𝛿𝑧₂−𝑧₁𝜉=−𝑔,₂−−𝜉=−ℎ,2𝛿𝛾𝑧₂−2𝛽𝛿1+𝑧₂−2𝜂𝛿1+𝑧₂²−2𝜉𝛿𝑧₂−𝑧₁𝛿=−𝜒,₂−𝛿=−𝜅.(2.24) For the fixed values of  𝑉₁,𝑢₁,𝛼,and𝑚, in particular, the system of ten equations (2.24) determines the sets of variables 𝐸_0,𝐽,𝑉₀,𝑉₂,𝑢₀,𝑢₂,𝛾,𝛽,𝜉,𝜂, and 𝛿. Therefore, the spectrum and eigenfunctions of the system are easily obtained for a particular system. For the higher states, the mathematical process is more cumbersome and complicated but can be followed by the same token we did here, that is, by choosing 𝑓₁(𝑧)=(𝑧−𝛼¹₁)for the first node,  𝑓₂(𝑧)=(𝑧−𝛼²₁)(𝑧−𝛼²₂)  for the second node, and so forth.

3. Conclusion

Motivation behind our study was the high number of spin-zero relativistic systems that we frequently face as well as the attractive structure of the Deng-Fan potential. The corresponding ordinary differential equation was too complicated to be solved by common analytical techniques. We, therefore, performed some novel transformations and applied an acceptable approximation to the centrifugal term. We next proposed an interesting physical ansatz solution by which we were able to find a quasi-analytical solution. Our results are particularly useful in particle and nuclear physics and can be directly used after prerequisite fits performed.

Acknowledgment

It is a great pleasure for the authors to thank the referee for his careful comments on the paper.