1. Introduction

The Klein-Gordon and Dirac equations, or relativistic mod equations, can be solved analytically or numerically to offer us a lot of information about a physical systems. Various techniques are available in the literature to obtain their solutions, such as the parametric Nikiforov-Uvarov (pNU) method, the exact quantization rule, the Qiang-Dong proper quantization rule, the path integral method, the asymptotic iteration method (AIM), the factorization method, the Laplace transform approach, the supersymmetric quantum mechanics (SUSYQM), the ansatz method, and the series expansion method. These equations play a vital role in statistical physics, solid-state physics, quantum field theory, atomic and sub-atomic physics, and molecular physics. Various research has been carried out on these equations of many considerable potentials. Parmar and Vinodkumar (2021) studied the combined modified Yukawa and Kratzer potential (MYKP) in the case of the Klein-Gordon equation (KGE) for calculated numerical results of the energy spectrum for CO, H2, LiH, and HCl molecules by pNU and SUSYQM methods using the Greene-Aldrich approximation to handle 1/r and 1/r ² terms in the effective potential [¹]. Many researchers have dealt with either the modified Yukawa or Kratzer potentials either individually or through the combination of one or both of them with other potentials within the framework of the different fundamental equations. Edet et al. analyzed the modified Kratzer potential plus the screened Coulomb potential by the pNU method and obtained the energy spectrum [²]. Ikot et al. obtained the exact bound state energy spectrum of the Schrödinger equation with energy-dependent molecular Kratzer potential using AIM [³]. Ikot et al. (2019) obtained the energy eigenvalues and the corresponding normalized eigenfunctions of screened Kratzer potential for lithium hydride (LiH) and hydrogen chloride (HCl) diatomic molecules within the framework of non-relativistic quantum mechanics via the pNU method. Ikot et al. (2019) obtained the energy eigenvalues and the corresponding normalized eigenfunctions of the screened Kratzer potential for lithium hydride (LiH) and hydrogen chloride (HCl) diatomic molecules within the framework of non-relativistic quantum mechanics via the pNU method [⁴]. Ahmadov et al. [⁵] studied solutions of KGE with the Manning-Rosen equation plus a class of Yukawa potentials using pNU and SUSYQM methods and presented the energy spectrum for any l-state and the corresponding radial wave functions in terms of the hypergeometric functions. Purohit et al. (2021) solved the Schrodinger¨ equation in D dimensions and obtained the eigenspectrum of the energy and momentum for time-independent and timedependent Hulthen-screened cosines Kratzér potentials Using the Qiang-Dong proper quantization rule and the supersymmetric quantum mechanics approach [⁶]. Recently, Purohit et al. (2022) obtained the energy spectrum for MYKP with the magnetic field and Aharanov-Bohm flux field using the pNU approach and SEM in this study [⁷]. The Kratzer potential (KP), which is known first and foremost by Kratzer itself [⁸], is used in quantum atomic-molecular physics and has played a vital role in the history of molecular and quantum chemistry [⁹]. Purohit et al. (2020) [¹⁰] obtained via the generalized Nikiforov-Uvarov method the approximate bound-state solutions of the D-dimensional KGE for screened cosine Kratzer potential (SCKP) using approximation suggested by Greene-Aldrich. Next year, Purohit et al. also investigated SCKP under the influence of the magnetic field and Aharanov-Bohm flux field and obtained energy eigenvalues and wave functions with external fields via the pNU method using the approximation method suggested by Greene-Aldrich for handling centrifugal barriers [¹¹]. On the other hand, Yukawa potential, often called screened Coulomb potential, is a short-range potential that has applications in particle, high-energy, and molecular physics. It is used to study the interaction that occurs between the atoms of diatomic molecules [¹²-¹⁴]. Recently, Purohit et al. (2022) used the linear plus modified Yukawa potential as the quarkantiquark interaction potential and obtained the energy eigenvalues and associated wave function by solving the KGE analytically using the pNU method [¹⁵]. Based on this motivation, in this work, we study new approximate bound state solutions of the deformed Klein-Gordon equation (DKGE) in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries of a newly proposed combined potential called the improved modified exponential screened plus Yukawa potential (IMYKP) within the framework of parametric Bopp’s shift method. This is a new potential model that has not yet been studied to the best of our knowledge in 3D-ERQM symmetries. The main objective of this study is to deepen the study of research [¹] performed in the frameworks of symmetries known in quantum mechanics known in the literature and research on new applications of this potential. It is worth noting that the first foundations of quantum mechanics were based on non-commutative axioms for all positions q^μ(s,h,i)*q^ν(s,h,i)≠q^ν(s,h,i)*q^μ(s,h,i) and momentum π^μ(s,h,i)*πνs,h,i ≠π^ν(s,h,i)*π^μ(s,h,i), individually or in combination q^μ(s,h,i)*π^ν(s,h,i)≠π^ν(s,h,i)*q^μ(s,h,i), here (∗) stands for the Weyl-Moyal star product. However, the scientific task was not accomplished due to mathematical difficulties. This made the specialized researchers ignore two basic axioms and complete the research path in the last axiom, and this is what is known as the relative and nonrelativistic quantum mechanics known in the literature. In recent years, real indicators have appeared, for example, the divergence problem in the standard model (SM), where gravitational forces are not included in addition to many other problems [¹⁶-²⁶]. It should be mentioned that before the renormalization approach was created and gained popularity, Heisenberg proposed the idea of extended non-commutativity to the coordinates as a possible treatment for eliminating the limitless numbers of field theories in 1930. Snyder published the first work on QFT’s history in 1947 [²⁷], and Connes introduced its geometric analysis in 1991 and 1994 [²⁸, ²⁹] in an effort to standardize QFT. Seiberg and Witten obtain a new version of gauge fields in noncommutative gauge theory [³⁰] by extending earlier ideas on the advent of NC geometry in string theory with a nonzero B-field. The elimination of the observed unwanted divergences or infinities that appear to produce short-range in field theories like gravitational theory by creating new quantum fluctuations is one of the potential objectives of NC deformation of space-space and phase-phase. The noncommutative R⁴extended to the supersymmetric field theories for the U(1) vector multiplets and chiral multiplets of the fundamental, anti-fundamental and adjoint representations of the gauge group [³¹], this shows another confirmation of the validity and effectiveness of the non-commutative theory. It is important to refer to some of our work related to the study of each of the two potentials under study in the framework of the extended Schrodinger¨ equation such as new modified Kratzér-type interactions [³²], modified Kratzer potential [³³] and the generalized perturbed Yukawa potential [³⁴], DKGE such as a linear combination of Hulthen and Kratzer potentials [³⁵], modified More general exponential screened Coulomb potential plus Yukawa potential [³⁶], generalized modified screened Coulomb plus inversely quadratic Yukawa potential [³⁷], Manning-Rosen plus quadratic Yukawa potential [³⁸], the deformed unequal scalar and vector Hellmann plus modified Kratzer potentials [³⁹], modified equal scalar and vector Manning-Rosen and Yukawa potentials [⁴⁰] and the Kratzer potential which studied by Darroodi et al. [⁴¹]. For the deformed Dirac equation, we have studied both modified Yukawa potential [⁴²], improved inversely quadratic Yukawa potential within improved Coulomb-like tensor interaction [⁴³], improved Schiöberg potential within the Yukawa tensor interaction, new modified Yukawa potential [⁴⁴], new modified Yukawa potential [⁴⁵], spatially dependent mass for the improved Eckart potential including the improved Yukawa tensor interaction [⁴⁶]. Furthermore, the modified Kratzer potential was studied in the framework of the Duffin-Kemmer-Petiau (DKP) equation [⁴⁷]. The following are the vector and scalar IMYKP models that will be used in this study (V _yk (rb) and S _yk (rb)) as follows:

where (V _yk (r),S _yk (r)) are the (vector, scalar) modified Yukawa-Kratzer potential, according to the view of 3D-RQM and 3D-NRQM symmetry, known in the literature [¹]:

and

where potential strength parameters A ₁ ≡ A ₃ ≡ V ₀, A ₂ ≡ 2V ₀, A ₄ ≡ 2D _e r _e , A5≡Dere2 and α is the screening parameters, D _e is the dissociation energy, r _e is the equilibrium bond length, (r^andr) are the interatomic distances in the extended QM and usual QM symmetries, respectively. The coupling LΘ is the scalar product of the usual components of the angular momentum operator L(L _x ,L _y ,L _z ) and the infinitesimal non-commutativity vector Θ (θ ₁₂ ,θ ₂₃ ,θ ₁₃)/2. In the case of G _NC , the noncentral generators can be suitably realized as self-adjoint differential operators (q^μ(s,h,i), π^ν(s,h,i)) appear in n three varieties. The first is the canonical structure (CS) variety, the second is the Lie structure (LS) variety, and the last corresponds to the quantum plane (QP) variety in the representations of Schrodinger, Heisenberg, and interaction pictures, satisfying a¨ deformed algebra of the form (for simplicity, we have used the natural units ℏ=c=1) [²⁵,⁴⁸-⁵⁵]:

and

with Λαβ≡q^α(s,h,i)q^β(s,h,i), here q^μ(s,h,i) can be equal one of (x^μs∨x^μh ∨x^μi) and π^μ(s,h,i) can be equal one of (p^μs ∨ p^μh ∨ p^μi) which are the generalized coordinates and the corresponding generalizing momentums, respectively, in 3D-EQM symmetries, while the corresponding coordinates xμ(s,h,i) (xμs ∨ xμh ∨ xμi) and pμs,h,i pμs ∨ pμh ∨ pμi in the usual 3D-QM symmetries, respectively, (∨≡or) Additionally, the uncertainty relation (in LHS of the below equation) that corresponds to the LHS of Eq. (3), will become in 3D-EQM symmetries as follows:

On the other hand, we notice that the RHS of Eq. (4) generates a novel uncertainty relation:

here h _µν and G _µν are equal to the average values:

The new subdivided three-uncertainties relations in Eq. (6) have no comparison in the existing literature. Under the Lorentz transformation, which includes boosts and/or rotations of the observer’s inertial frame, Eqs. (3) and (4) are covariant equations (have the same behavior as q^μ(s,h,i). We are expanded the modified equal-time noncommutative canonical commutation relations (METNCCCRs) to include both Heisenberg and interaction pictures in both 3D-ERQM and 3D-ENRQM symmetries. Here δ _µν is the Kronecker symbol, (µ,ν = 1,2,3), θ _µν is antisymmetric real constant (3 × 3) matrices with the dimensionality (length)² parameterizing the deformation of space-space, ϵ _µν is an antisymmetric tensor operator describing the non-commutativity of space-time (ϵ _µν = − ϵ _µν = 1 for µ ≠ ν and ϵ _ϵϵ = 0) and θ ∈ R is the noncommutative parameter, ℏeff≅ℏ is the effective Planck constant. The new deformed scalar product h(x) ∗ g(x) is defined by the Weyl-Moyal ∗-product in three different ways [⁵⁶-⁶¹]:

The second component in Eq. (7) provides a physical representation of the consequences of space-space non-commutativity. The outline of the paper is as follows: Section 2 presents an overview of the 3D-KGE under the modified Yukawa-Kratzer potential models. Section 3 is devoted to investigating the 3D-DKGE using the well-known Bopp’s shift method to obtain the effective potential for the IMYKP model. Furthermore, using standard perturbation theory, we find the expectation values of the radial terms (1/(1 − z)⁴, z ² /(1 − z)³, z ² /(1 − z)⁴, z/(1 − z)², z/(1 − z)³ and 1/(1 − z)³) to calculate the corrected relativistic energy generated by the effect of the perturbed effective potential Ξpertyk(r) of the IMYKP model, and we derive the global corrected energies for bosonic particles and antiparticles whose spin quantum number has an integer value. Section 4 is reserved to analyze the non-relativistic limit, in 3D-ENRQM symmetries for the IMYKP model. In the next section, we will discuss the most important special cases in the relativistic and non-relativistic cases, which are useful for specialists and readers alike. Section 6 is reserved for the study of spin-averaged mass spectra of HLM under IMKSCP models in 3D-ENRQM symmetries. Finally, we present our conclusion in Sec. 7.

3. The new solutions of DKGE under the IMYKP models in 3D-ERQM symmetries: